10.2: Double Integrals over General Regions
- Page ID
- 144351
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \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 the volume under the surface \(z = 3xy\) and under the region \(R\) bounded by \(y = x^3\), \(y = x^3 + 1\), \(x = 0\), and \(x = 1\).
- Find the volume under the surface \(z = xy + 1\) and under the region \(R\) bounded by \(x = y^2 - 1\) and \(x = \sqrt{1 - y^2}\).
- Find the volume bounded by \(y = \sqrt{4x}\), \(2x + y = 4\), \(z = y\), \(y = 0\), and \(z = 0\).
- Find the volume in the first octant bounded by \(y^2 = 4 - x\) and \(y = 2z\).
- Find the volume in the first octant bounded by \(x + y + z = 9\), \(2x + 3y = 18\), and \(x + 3y = 9\).
- Find the volume in the first octant bounded by \(x^2 + y^2 = a^2\) and \(z = x + y\).
- Evaluate \(\displaystyle \iint_R x^2\ dA\), where \(R\) is the region in the first quadrant bounded by \(xy = 16\) and the lines \(y = x\), \(y=0\), and \(x = 8\).
- Evaluate \(\displaystyle \iint_R \ dA\), where \(R\) is the region bounded by \(x = \dfrac{\pi}{2}\), \(x = y - 1\), and \(y = \cos x\).
- Evaluate \(\displaystyle \iint_R xy\ dA\), where \(R\) is the triangular region with vertices \((0, 0)\), \((0, 2)\), and \((2, 2)\).
- Use a double integral to find the area of region \(R\), where \(R\) is the triangular region with vertices \((0, 0)\), \((-3, 5)\), and \((4, 2)\).
- Use a double integral to find the area of region \(R\), where \(R\) is the region inside the circle \((x-1)^2 + y^2 = 2\) outside the circle \(x^2 + y^2 = 1\).
- Reverse the order of integration for \(\displaystyle \int_{-1}^{\pi/2} \int_{0}^{x+1} f(x,y)\ dy\,dx\).
- Reverse the order of integration for \(\displaystyle \int_{0}^{1} \int_{x-1}^{1-x} f(x,y)\ dy\,dx\).
- Reverse the order of integration for \(\displaystyle \int_{-1}^{0} \int_{-\sqrt{y+1}}^{\sqrt{y+1}} f(x,y)\ dx\,dy\).
- Reverse the order of integration for \(\displaystyle \int_{-1/2}^{1/2} \int_{-\sqrt{y^2+1}}^{\sqrt{y^2+1}} f(x,y)\ dx\,dy\).
- Reverse the order of integration for \(\displaystyle \int_{1}^{2} \int_{0}^{\ln x} f(x, y)\ dy\,dx\).
- Reverse the order of integration for \(\displaystyle \int_{0}^{1} \int_{4x}^{4} f(x, y)\ dy\,dx\).
- Reverse the order of integration for \(\displaystyle \int_{0}^{3} \int_{0}^{\sqrt{9 - y^2}} f(x, y)\ dx\,dy\).