Question

Using polar coordinates, evaluate the integral ∫∫R sin(x^2+y^2)dA where R is the region 4≤x^2+y^2≤36.

Answer 1

Let's draw this region:

Let's move on to polar coordinates

$x = r\cos \varphi ,\,\,y = r\sin \varphi ,\,\,0 < \varphi < 2\pi ,\,\,2 < r < 6$

Then

$\begin{array}{l} \int\limits_R {\int {\sin \left( {{x^2} + {y^2}} \right)} } dA = \int\limits_0^{2\pi } {d\varphi } \int\limits_2^6 {r\sin {r^2}dr} = \frac{1}{2}\int\limits_0^{2\pi } {d\varphi } \int\limits_2^6 {\sin {r^2}d{r^2}} = - \frac{1}{2}\left. \varphi \right|_0^{2\pi }\left. {\cos {r^2}} \right|_2^6 = - \pi (\cos 36 - \cos 4) = \\ = \pi (\cos 4 - \cos 36) \end{array}$