Using polar coordinates evaluate the double integral of $\sin(x^2 + y^2) \mathrm{d}A$,
where the region is $4 \le x^2 + y^2 \le 64$.
I know that we have to find the range of angle and the range of $r$. Should the angle be between $0$ and $2\pi$ and the $r$ between $2$ and $8$? And then how do I set up the integral?
Thank you.
Your question "how do I set up the integral?" is probably the most frequently asked question in my multivariable calculus course.
A double integral in $\mathbb{R}^2$, $\displaystyle \iint_D f(x,y)dA$, has three pieces of data: the region of integration $D$, the function $f(x,y)$, and the area differential $dA$. Setting up the integral in polar coordinates requires you to transform/convert all three things: $dA \to r dr d\theta$, $f(x,y) \to f(r\cos \theta, r \sin \theta)$, and describing $D$ in polar coordinates.
A nice shortcut is that for polar coordinates $x = r \cos \theta$ and $y = r \sin \theta$ the expression $x^2 + y^2$ always transforms to $r^2$, a consequence of the Pythagorean identity.