11.6: Surface Integrals
- Page ID
- 144363
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Find a parametric description for the surface \(3x - 2y + z = 2\).
- Find a parametric description for the surface \(z = x^2 + y^2\).
- Find a parametric description for the surface \(z^2 = x^2 + y^2\) for \(2 \leq z \leq 8\).
- Find a parametric description for the surface \(x^2 + y^2 = 9\).
- Find a parametric description for the surface \(x^2 + y^2 + z^2 = 4\).
- Find a parametric description for the portion of the sphere \(x^2 + y^2 + z^2 = 4\) below the plane \(z = 1\).
- Let \(\sigma\) be the portion of the sphere \(x^2 + y^2 + z^2 = 4\) with \(z \geq 0\). Evaluate \(\displaystyle \iint_\sigma z\ dS\).
- Let \(\sigma\) be the portion of the cone \(z = \sqrt{x^2 + y^2}\) between \(z = 1\) and \(z = 3\). Evaluate \(\displaystyle \iint_\sigma z^2\ dS\).
- Let \(\sigma\) be the portion of the plane \(x + y + z = 1\) in the first octant. Evaluate \(\displaystyle \iint_\sigma xy\ dS\).
- Evaluate \(\displaystyle \iint_\sigma (2, -3, 4) \cdot \mathbf{N}\ dS\), where \(\sigma\) is the paraboloid \(z = x^2 + y^2\) with \(-1 \leq x \leq 1\), \(-1 \leq y \leq 1\), oriented up.
- Evaluate \(\displaystyle \iint_\sigma (x, y, -2) \cdot d\mathbf{S}\), where \(\sigma\) is given by \(z = 1 - x^2 - y^2\), where \(x^2 + y^2 \leq 1\), oriented up.
- Evaluate \(\displaystyle \iint_\sigma (xy, yz, zx) \cdot d\mathbf{S}\), where \(\sigma\) is the paraboloid \(z = x + y^2 + 2\), where \(0 \leq x \leq 1\), \(x \leq y \leq 1\), oriented up.