polar coordinates question

Question

I was tasked with writing $\iint_D f(x,y) \,dx \,dy$ for $ [ D:{4\leq x^2 + y^2 \leq}9]$ through ''reoccurring integrals'' in polar and Cartesian systems? what are ''reoccurring integrals''? and how can you do it in both systems?

Answer 1

Write it as two iterated integrals. In polar coordinates, notice that $$D = \{(r,\theta) \mid 4 \leq r^2 \leq 9, 0 \leq \theta \leq 2\pi \} = \{ (r,\theta) \mid 2 \leq r \leq 3, 0 \leq \theta \leq 2\pi \}.$$ Your set $D$ is an annulus in cartesian coordinates, but is is a rectangle in polar coordinates. Remembering that ${\rm d}x\,{\rm d}y = r\,{\rm d}r\,{\rm d}\theta$, you can write: $$\int_0^{2\pi}\int_2^3 r\,f(r,\theta)\,{\rm d}r\,{\rm d}\theta.$$ Since circles don't define functions, you would have to break your integral into two or more integrals to write it in cartesian coordinates (we don't have any information about $f$, about its symmetries, etc)