극좌표 이중적분

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, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates.

$\int_{1}^{2}\,\int_{0}^{x}({x}^{2}+{y}^{2})dy\,dx=\int_{0}^{\frac{\pi }{4}}\,\int_{\sec \,\theta }^{2\,\sec \,\theta }{r}^{3}dr\,d\theta$

In the following exercises, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates.

$\int_{2}^{3}\,\int_{0}^{x}\frac{x}{\sqrt{{x}^{2}+{y}^{2}}}dy\,dx=\int_{0}^{\pi /4}\,\int_{2\,\sec \,\theta }^{3\sec \,\theta }r\,\cos \,\theta \,dr\,d\theta$

In the following exercises, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates.

$\int_{0}^{1}\,\int_{{x}^{2}}^{x}\frac{1}{\sqrt{{x}^{2}+{y}^{2}}}dy\,dx=\int_{0}^{\pi /4}\,\int_{0}^{\tan \,\theta \,\sec \,\theta }dr\,d\theta$

In the following exercises, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates.

$\int_{0}^{1}\,\int_{{x}^{2}}^{x}\frac{y}{\sqrt{{x}^{2}+{y}^{2}}}dy\,dx=\int_{0}^{\pi /4}\,\int_{0}^{\tan \,\theta \,\sec \,\theta }r\,\sin \,\theta \,dr\,d\theta$

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 $r=1+\cos \,\theta .$

Find the area of the region $D$ bounded by the polar axis and the upper half of the cardioid $r=1+\cos \,\theta .$

Evaluate the integral $\iint_{D}\,dA,$ where $D$ is the region bounded by the part of the four-leaved rose $r=\sin \,2\theta$ situated in the first quadrant (see the following figure).

Find the total area of the region enclosed by the four-leaved rose $r=\sin \,2\theta$ (see the figure in the previous exercise).

Find the area of the region $D,$ which is the region bounded by $y=\sqrt{4-{x}^{2}},$ $x=\sqrt{3},$ $x=2,$ and $y=0.$

Find the area of the region $D,$ which is the region inside the disk ${x}^{2}+{y}^{2}\le 4$ and to the right of the line $x=1.$

Determine the average value of the function $f(x,y)={x}^{2}+{y}^{2}$ over the region $D$ bounded by the polar curve $r=\cos \,2\theta ,$ where $-\frac{\pi }{4}\le \theta \le \frac{\pi }{4}$ (see the following graph).

Determine the average value of the function $f(x,y)=\sqrt{{x}^{2}+{y}^{2}}$ over the region $D$ bounded by the polar curve $r=3\,\sin \,2\theta ,$ where $0\le \theta \le \frac{\pi }{2}$ (see the following graph).

Find the volume of the solid situated in the first octant and bounded by the paraboloid $z=1-4{x}^{2}-4{y}^{2}$ and the planes $x=0,y=0,$ and $z=0.$

Find the volume of the solid bounded by the paraboloid $z=2-9{x}^{2}-9{y}^{2}$ and the plane $z=1.$