Double Integrals In Polar Coordinates

(requires JavaScript)

1. For each region shown, decide whether to use polar or rectangular coordinates and write the iterated integral $\underset{R}{\iint }f<!--FUNCTION\; APPLICATION-->\left(x,y\right)dA$, where $f$ is an arbitrary continuous function.

   a. b.

   c. d.

   e. f.

2. Sketch the region whose area is given by the integral and evaluate the integral

   ${\displaystyle {\int }_0^{\pi /2}{\int }_0^{4\cos <!--FUNCTION\; APPLICATION-->\left(\theta \right)}rdrd\theta }$

3. Evaluate the integral

   ${\displaystyle \underset{R}{\iint }\cos <!--FUNCTION\; APPLICATION-->\left(x^2+y^2\right)dA}$

   by changing to polar coordinates. Here $R$ is the region that lies above the $x\text{-axis}$ within the circle $x^2+y^2=9$.

   ${\displaystyle \frac{1}{2}\pi \sin <!--FUNCTION\; APPLICATION-->\left(9\right)}$

4. Evaluate the integral

   ${\displaystyle \underset{R}{\iint }{\tan }^{-1}<!--FUNCTION\; APPLICATION-->\left(y/x\right)dA}$

   by changing to polar coordinates, given that

   $R=\left\{\left(x,y\right)|1\le x^2+y^2\le 4,0\le y\le x\right\}$

   ${\displaystyle \frac{3}{64}{\pi }^2}$
5. Use polar coordinates to find the volume of the solid above the cone $z=\sqrt{x^2+y^2}$ and below the sphere $x^2+y^2+z^2=1$.
   ${\displaystyle \frac{2\pi }{3}\left(1-\frac{1}{\sqrt{2}}\right)}$
6. Use polar coordinates to find the volume of the solid inside both the cylinder $x^2+y^2=4$ and the ellipsoid $4x^2+4y^2+z^2=64$.
   ${\displaystyle \frac{8\pi }{3}\left(64-24\sqrt{3}\right)}$
7. Evaluate the iterated integral ${\displaystyle {\int }_{-3}^3{\int }_0^{\sqrt{9-x^2}}\sin <!--FUNCTION\; APPLICATION-->\left(x^2+y^2\right)dydx}$ by converting to polar coordinates.
   ${\displaystyle \frac{1}{2}\pi \left(1-\cos <!--FUNCTION\; APPLICATION-->\left(9\right)\right)}$
8. Evaluate the iterated integral ${\displaystyle {\int }_0^2{\int }_0^{\sqrt{2x-x^2}}\sqrt{x^2+y^2}dydx}$ by converting to polar coordinates.
   ${\displaystyle \frac{16}{9}}$