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  1. Evaluate \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\iint_S xy\:dA where \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}S is the disk of radius 3 centered at the origin.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^{2\pi}\int_0^3 r^2\sin\theta\:\cos\theta\:drd\theta

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\sin\theta\:\cos\theta\:\int_0^3 r^2 dr
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {1\over3}\sin\theta\:\cos\theta\:[r^3|_{r=0}^3
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 3\sin\theta\:\cos\theta

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~ u = \sin\theta
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~ du = \cos\theta\:d\theta

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}3\int_0^{2\pi} \sin\theta\:\cos\theta\:d\theta
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {3\over2}[\sin^2\theta|_{\theta=0}^{2\pi}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {3\over2}(0-0)
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 0

  2. Evaluate \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\iint_S (x - y)dA where \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}S is the region bounded by \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}x^2 + y^2 = 1 and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}x^2 + y^2 = 4.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^{2\pi}\int_1^2 r\cos\theta\:drd\theta + \int_0^{2\pi}\int_1^2 r\sin\theta\:drd\theta

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}k(\theta)\int_1^2 r\:dr
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {1\over2}k(\theta)[r^2|_{r=1}^2
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {3\over2}k(\theta)

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}{3\over2}\int_0^{2\pi}\cos\theta\:d\theta + {3\over2}\int_0^{2\pi}\sin\theta\:d\theta
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {3\over2}[\sin\theta|_0^{2\pi} - {3\over2}[\cos\theta|_0^{2\pi}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {3\over2}(0-0) - {3\over2}(1-1)
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 0

  3. Evaluate \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\iint_S \sqrt{9-x^2-y^2}\:dA where \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}S is the region bounded by \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}x^2 + y^2 \le 4 and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}x \ge 0.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^\pi\int_0^2 \sqrt{9 - r^2(\cos^2\theta + \sin^2\theta)}drd\theta
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \int_0^\pi\int_0^2 \sqrt{9 - r^2}drd\theta

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^2 \sqrt{9-r^2}dr

  4. Find the volume of the solid enclosed by the paraboloid \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}z = 18 - 2(x^2+y^2) and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}z \ge 0.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}S:~ {(x,y)~|~ 0 \le 18 - 2(x^2 + y^2)}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {(x,y)~|~ x^2 + y^2 \le 9}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {(r,\theta)~|~ r \le 3 ~\land~ 0 \le \theta \le 2\pi}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^{2\pi}\int_0^3 (18 - 2r^2(cos^2\theta + \sin^2\theta))drd\theta
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \int_0^{2\pi}\int_0^3 (18 - 2r^2)drd\theta

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^3 (18 - 2r^2)dr
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 18[r|{r=0}^3 - {2\over3}[r^3|^3
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 54 - 18 \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 36

  5. Evaluate \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\iiint_S \sqrt{x^2+y^2}\:dV where \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}S is the region bounded by the cylinder \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}x^2+y^2=16, z=-5, z=4.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}S:~ {(x,y,z)~|~ x^2+y^2 \le 16 ~\land~ -5 \le z \le 4}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {(\rho,\phi,z)~|~ \rho \le 4 ~\land~ 0 \le \phi \le 2\pi ~\land~ -5 \le z \le 4}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_{-5}^4\int_0^{2\pi}\int_0^4 \sqrt{r^2\cos^2\theta + r^2\sin^2\theta}\:d\rho d\phi dz
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \int_{-5}^4\int_0^{2\pi}\int_0^4 r\:d\rho d\phi dz
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {1\over2}(16-0)\int_{-5}^4\int_0^{2\pi} d\phi dz
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 8\int_{-5}^4\int_0^{2\pi} d\phi dz
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 16\pi\int_{-5}^4 dz \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 144\pi

  6. Evaluate \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\iiint_S x^2 dV where \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}S is the region inside the cylinder \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}x^2 + y^2 = 1, below \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}z=0, and above the cone \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}z^2 = 4x^2 + 4y^2.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}S:~ {(x,y,z)~|~ x^2+y^2 \le 1 ~\land~ -2\sqrt{x^2+y^2} \le z \le 0}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {(\rho,\phi,z)~|~ \rho \le 1 ~\land~ 0 \le \theta \le 2\pi ~\land~ -2\rho \le z \le 0}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^{2\pi}\int_0^1\int_{-2\rho}^0 \rho\cos\phi\:dz d\rho d\phi

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\rho\cos\phi\:\int_{-2\rho}^0 dz
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 2\rho^2\cos\phi

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}2\cos\phi\:\int_0^1\rho^2 d\rho
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {2\over3}\cos\phi\:[\rho^3|_{\rho=0}^1
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {2\over3}\cos\phi

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}{2\over3}\int_0^{2\pi} \cos\phi\:d\phi
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {2\over3}[\sin\phi|_{\phi=0}^2\pi
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {2\over3}(0-0) \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 0

  7. Find the mass of the solid bounded by the paraboloid \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}z = 4x^2 + 4y^2 and the plane \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}z=4 if the solid has constant density \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}k.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}S:~ {(\rho,\phi,z)~|~ \rho \le 1 ~\land~ 0 \le \phi \le 2\pi ~\land~ 4\rho^2 \le z \le 4}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^{2\pi}\int_0^1\int_{4\rho^2}^4 k\:dz d\rho d\phi

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}k\int_{4\rho^2}^4 dz
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 4k - 4k\rho^2

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}4k\int_0^1 d\rho - 4k\int_0^1 \rho^2 d\rho
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 4k - {4\over3}k \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {8\over3}k

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}{8\over3}k\int_0^{2\pi} d\phi
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {16\pi\over3}k

  8. Evaluate \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\iiint_S (x^2 + y^2 + z^2)^2 dV where \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}S is the sphere of radius 5 centered at the origin.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^{2\pi}\int_0^\pi\int_0^5 r^4 dr d\theta d\phi

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^5 r^4 dr
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {1\over5}(5^4) \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 125

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}125\int_0^{2\pi}\int_0^\pi d\theta d\phi
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 250\pi^2

  9. \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\iiint_S (16 - x^2 - y^2)dV where \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}S is the solid upper hemisphere \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}x^2+y^2+z^2 \le 16, z \ge 0.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^{2\pi}\int_0^{\pi/2}\int_0^4 (16 - r^2\sin^2\theta\:(\cos^2\phi + \sin^2\phi))dr d\theta d\phi
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \int_0^{2\pi}\int_0^{\pi/2}\int_0^4 (16 - r^2\sin^2\theta)dr d\theta d\phi

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^4 (16 - r^2\sin^2\theta)dr
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 64 - {1\over3}\sin^2\theta\:(4^3-0)
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 64 - {64\over3}\sin^2\theta

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^{\pi/2} (64 - {64\over3}\sin^2\theta)d\theta
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \int_0^{\pi/2} (64 - {32\over3}(1 - \cos(2\theta)))d\theta
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \int_0^{\pi/2} (64 - {32\over3} + {32\over3}\cos(2\theta))d\theta
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 32\pi-{16\pi\over3}+ {16\over3}[\cos(2\theta)|_{\theta=0}^{\pi/2}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 32\pi - {16\pi\over3} + {16\over3}(-1-1) \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {80\pi - 32 \over 3}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}{80\pi - 32 \over 3}\int_0^{2\pi} d\phi
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= {160\pi^2 - 64\pi \over 3}

  10. Find the mass of a solid sphere of radius \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}R is the density of the solid is proportional to its radial distance, i.e. if \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}f = kr for some constant \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}k.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\int_0^{2\pi}\int_0^\pi\int_0^R kr\:dr d\theta d\phi
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \int_0^{2\pi}\int_0^\pi {1\over2}kR^2 d\theta d\phi
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \pi^2kR^2

  11. Suppose \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v} = \vc{5}{-12}{+0} and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{w} = \vc{-3}{-6}{+0}. Plot these vectors in the xy plane (which you can do since the z components of these vectors are 0). Find the following: \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}-2\vv{v}+5\vv{w}, |\vv{v}|, |\vv{w}|, \vh{v}, \vh{w}, \vv{v}\cdot\vv{w}, and the (approximate) angle between \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v} and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{w}.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}-2\vv{v} + 5\vv{w} = \vc{-10}{+24}{+0} + \vc{-15}{-30}{+0}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \vc{-25}{-6}{+0}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}|\vv{v}| = \sqrt{5^2 + 12^2}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \sqrt{25 + 144} = \sqrt{169} = 13

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}|\vv{w}| = \sqrt{3^2 + 6^2}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vh{v} = \vc{{5\over13}}{-{12\over13}}{+0}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vh{w} = \vc{-{1\over\sqrt{5}}}{-{2\over\sqrt{5}}}{+0}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}\cdot\vv{w} = -15 + 72 = 57

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\cos\theta = {57\over39\sqrt{5}}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\theta = \arccos{57\over39\sqrt{5}}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~\approx 0.858


© Emberlynn McKinney