10.2: Double Integrals over General Regions
- Page ID
- 144351
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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 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 \(\displaystyle \int_{0}^{3} \int_{0}^{\sqrt{9 - y^2}} f(x, y)\ dx\,dy\).