10.3: Double Integrals in Polar Coordinates
- Page ID
- 144352
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Let \(R \subset \mathbb{R}^2\) be the portion of the disk of radius \(4\) centered at the origin that lies in the first quadrant. Express region \(R\) in polar coordinates.
- Let \(R \subset \mathbb{R}^2\) be the annulus that lies between the circles of radius \(1\) and radius \(2\). Express region \(R\) in polar coordinates.
- Let \(R \subset \mathbb{R}^2\) be the region in the first quadrant bounded by \(x^2 + y^2 = 1\), \(x^2 + y^2 = 9\), \(y = x\), and \(y = 0\). Express region \(R\) in polar coordinates.
- Let \(R \subset \mathbb{R}^2\) be the region bounded by \(y = x\), \(y = 0\), \(x = 1\), and \(x = 5\). Express region \(R\) in polar coordinates.
- Let \(R \subset \mathbb{R}^2\) be the region bounded by \(y = x\) and \(y = x^2\). Express region \(R\) in polar coordinates.
- Use polar coordinates to evaluate \(\displaystyle \iint_R (x^2 + y^2)\ dA\) over \(R = \{(r, \theta)\ |\ 3 \le r \le 5,\ 0 \le \theta \le 2\pi\}\).
- Use polar coordinates to evaluate \(\displaystyle \iint_R \sqrt[3]{x^2 + y^2}\ dA\) over \(R = \left\{(r, \theta)\ \big|\ 0 \le r \le 1,\ \dfrac{\pi}{2} \le \theta \le \pi\right\}\).
- Use polar coordinates to evaluate \(\displaystyle \iint_R \arctan\left(\dfrac{y}{x}\right)\ dA\) over \(R = \left\{(r, \theta)\ \big|\ 1 \le r \le 2,\ \dfrac{\pi}{4} \le \theta \le \dfrac{\pi}{3}\right\}\).
- Evaluate \(\displaystyle \int_0^3 \int_0^\sqrt{9-y^2} (x^2 + y^2)\ dx\,dy\) by converting to polar coordinates.
- Evaluate \(\displaystyle \int_0^4 \int_{-\sqrt{16 - x^2}}^\sqrt{16 - x^2} \sin(x^2 + y^2)\ dy\,dx\) by converting to polar coordinates.
- Evaluate \(\displaystyle \int_{-4}^4 \int_{-\sqrt{16 - y^2}}^0 (2y - x)\ dx\,dy\) by converting to polar coordinates.
- Evaluate \(\displaystyle \int_0^2 \int_y^\sqrt{8-y^2} (x+ y)\ dx\,dy\) by converting to polar coordinates.
- Evaluate \(\displaystyle \int_0^a \int_{-\sqrt{a^2 - x^2}}^0 x^2y\ dy\,dx\) by converting to polar coordinates.
- Evaluate \(\displaystyle \int_0^1 \int_{x^2}^x \dfrac{1}{\sqrt{x^2 + y^2}}\ dy\,dx\) by converting to polar coordinates.