극좌표 이중적분

$D$ is the region of the disk of radius $2$ centered at the origin that lies in the first quadrant.

$D$ is the region bounded by the $y$-axis and $x=\sqrt{1-{y}^{2}}.$

$D$ is the region bounded by the $x$-axis and $y=\sqrt{2-{x}^{2}}.$

In the following exercises, express the region $D$ in polar coordinates.

$D=\{(x,y)|{x}^{2}+{y}^{2}\le 4x\}$

In the following exercises, express the region $D$ in polar coordinates.

$D=\{(x,y)|{x}^{2}+{y}^{2}\le 4y\}$

In the following exercises, evaluate the double integral $\iint_{R}f(x,y)dA$ over the polar rectangular region $D.$

$f(x,y)={x}^{2}+{y}^{2},D=\{(r,\theta )|3\le r\le 5,0\le \theta \le 2\pi \}$

In the following exercises, evaluate the double integral $\iint_{R}f(x,y)dA$ over the polar rectangular region $D.$

$f(x,y)=x+y,\,\,D=\{(r,\theta )|3\le r\le 5,0\le \theta \le 2\pi \}$

In the following exercises, evaluate the double integral $\iint_{R}f(x,y)dA$ over the polar rectangular region $D.$

$f(x,y)={x}^{2}+xy,D=\{(r,\theta )|1\le r\le 2,\pi \le \theta \le 2\pi \}$

In the following exercises, evaluate the double integral $\iint_{R}f(x,y)dA$ over the polar rectangular region $D.$

$f(x,y)={x}^{4}+{y}^{4},D=\{(r,\theta )|1\le r\le 2,\frac{3\pi }{2}\le \theta \le 2\pi \}$

In the following exercises, evaluate the double integral $\iint_{R}f(x,y)dA$ over the polar rectangular region $D.$

$f(x,y)=\sqrt[3]{{x}^{2}+{y}^{2}},$ where $D=\{(r,\theta )|0\le r\le 1,\frac{\pi }{2}\le \theta \le \pi \}.$

In the following exercises, evaluate the double integral $\iint_{R}f(x,y)dA$ over the polar rectangular region $D.$

$f(x,y)={x}^{4}+2{x}^{2}{y}^{2}+{y}^{4},$ where $D=\{(r,\theta )|3\le r\le 4,\frac{\pi }{3}\le \theta \le \frac{2\pi }{3}\}.$

In the following exercises, evaluate the double integral $\iint_{R}f(x,y)dA$ over the polar rectangular region $D.$

$f(x,y)=\sin \left(\arctan \,\frac{y}{x}\right),$ where $D=\{(r,\theta )|1\le r\le 2,\frac{\pi }{6}\le \theta \le \frac{\pi }{3}\}$

In the following exercises, evaluate the double integral $\iint_{R}f(x,y)dA$ over the polar rectangular region $D.$

$f(x,y)=\arctan \left(\frac{y}{x}\right),$ where $D=\{(r,\theta )|2\le r\le 3,\frac{\pi }{4}\le \theta \le \frac{\pi }{3}\}$

In the following exercises, evaluate the double integral $\iint_{R}f(x,y)dA$ over the polar rectangular region $D.$

$\iint_{D}{e}^{{x}^{2}+{y}^{2}}\left[1+2\,\arctan \left(\frac{y}{x}\right)\right]dA,\,D=\{(r,\theta )|1\le r\le 2,\frac{\pi }{6}\le \theta \le \frac{\pi }{3}\}$

In the following exercises, evaluate the double integral $\iint_{R}f(x,y)dA$ over the polar rectangular region $D.$

$\iint_{D}({e}^{{x}^{2}+{y}^{2}}+{x}^{4}+2{x}^{2}{y}^{2}+{y}^{4})\arctan \left(\frac{y}{x}\right)dA,\,D=\{(r,\theta )|1\le r\le 2,\frac{\pi }{4}\le \theta \le \frac{\pi }{3}\}$

In the following exercises, convert the integrals to polar coordinates and evaluate them.

$\int_{0}^{3}\,\int_{0}^{\sqrt{9-{y}^{2}}}({x}^{2}+{y}^{2})dx\,dy$

In the following exercises, convert the integrals to polar coordinates and evaluate them.

$\int_{0}^{2}\,\int_{-\sqrt{4-{y}^{2}}}^{\sqrt{4-{y}^{2}}}{({x}^{2}+{y}^{2})}^{2}dx\,dy$

In the following exercises, convert the integrals to polar coordinates and evaluate them.

$\int_{0}^{1}\,\int_{0}^{\sqrt{1-{x}^{2}}}(x+y)dy\,dx$

In the following exercises, convert the integrals to polar coordinates and evaluate them.

$\int_{0}^{4}\,\int_{-\sqrt{16-{x}^{2}}}^{\sqrt{16-{x}^{2}}}\sin ({x}^{2}+{y}^{2})dy\,dx$

Evaluate the integral $\iint_{D}r\,dA$ where $D$ is the region bounded by the polar axis and the upper half of the cardioid $r=1+\cos \,\theta .$