10.4: Triple Integrals
- Page ID
- 144353
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
- Evaluate \(\displaystyle \iiint_B (2x + 3y^2 + 4z^3)\ dV\), where \(B = [0, 1] \times [0, 2] \times [0, 3]\).
- Evaluate \(\displaystyle \iiint_B (z\sin x + y^2)\ dV\), where \(B = [0, \pi] \times [0, 1] \times [-1, 2]\).
- Evaluate \(\displaystyle \iiint_E (2x + 5y - 7z)\ dV\), where \(E\) is the region in the first octant bounded by \(y = 1-x\), \(z = 1\), and \(z = 2\)
- Evaluate \(\displaystyle \iiint_E (y \ln x + z)\ dV\), where \(E\) is the region in the first octant bounded by \(y = \ln x\), \(x = 1\), \(x = e\), and \(z = 1\).
- Evaluate \(\displaystyle \iiint_E (x + 2yz)\ dV\), where \(E\) is the region in the first octant where \(x + y + z \leq 5\), \(y \leq x\), and \(x \leq 1\).
- Evaluate \(\displaystyle \iiint_E y\ dV\), where \(E\) is the unit sphere centered at the origin.
- Evaluate \(\displaystyle \iiint_E x^2\ dV\), where \(E\) is the region bounded by \(x = 1 - y^2\), \(x = y^2 - 1\), \(z = 1\), and \(z = 2\).
- Evaluate \(\displaystyle \iiint_E 2x\ dV\), where \(E\) is the region in the first octant bounded by \(x + y + z = 4\), \(y = 2x\), and \(x = 2\).
- Use a triple integral to find the volume of the region in the first octant bounded by the paraboloid \(z = 1 - x^2 - y^2\) and the planes \(y = x\) and \(x = 0\)