11.6: Surface Integrals
- Page ID
- 144363
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
- 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.