Question

Evaluate the given integral by changing to polar coordinates. sin(x2 + y2) dA R , where R is the region in the first quadrant between the circles with center the origin and radii 3 and 5

Answer 1

In polar coordinates, the region $R$ is the set of points

$\left\{(r,\theta)\mid3\le r\le5,\,0\le\theta\le\dfrac\pi2\right\}$

and we have $x^2+y^2=r^2$ and $\mathrm dA=r\,\mathrm dr\,\mathrm d\theta$. So the integral, converted to polar, is

$\displaystyle\iint_R\sin(x^2+y^2)\,\mathrm dA=\int_0^{\pi/2}\int_3^5r\sin r^2\,\mathrm dr\,\mathrm d\theta=\frac\pi2\int_3^5r\sin r^2\,\mathrm dr$

Substitute $s=r^2$ to get

$=\displaystyle\frac\pi4\int_9^{25}\sin s\,\mathrm ds=\frac{(\cos9-\cos25)\pi}4$