10.2: Double Integrals over General Regions

Last updated

Mar 10, 2025
Save as PDF
- 10.1: Double Integrals over Rectangular Regions
- 10.3: Double Integrals in Polar Coordinates

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

Find the volume under the surface $z = 3xy$ and over 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 over 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 $\displaystyle \int_{0}^{3} \int_{0}^{\sqrt{9 - y^2}} f(x, y)\ dx\,dy$ .

Search

Text Color

Text Size

Margin Size

Font Type

Support Center

How can we help?