I am trying to find the volume underneath $f(x,y) = xy$ on the region $R$ shown in the picture.
I tried integral on polar coordinates:
$$\int_0^{\pi/2}\int_1^4 \left(r\cos\theta\right)\left(r\sin\theta\right)\, dr \, d\theta$$
but this doesn't give one of the answer choices so I don't know what is wrong.
I tried also multiplying by one of the random things you have to add on the end of polar coordinate integrals like $r$ or $r^2$ but it doesn't work
$$\int_0^{\pi/2}\int_1^4 \left(r\cos\theta\right)\left(r\sin\theta\right)r \, dr \, d\theta$$
and this doesn't work either
$$\int_0^{\pi/2}\int_1^4 \left(r\cos\theta\right)\left(r\sin\theta\right)r^2 \, dr \, d\theta$$
None of them give answers that are one of the four choices.
The correct one is $$ \int_0^{\pi/2}\int_1^4 \left(r\cos\theta\right)\left(r\sin\theta\right)r \, dr \, d\theta = \frac{255}{8} $$
where $rdrd\theta$ is the differential area element and $(r\cos\theta)(r\sin\theta)$ is the height of the differential area element.