 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-2/18/a/solution

The integrand separates into a radial and an angular factor, so

\int_{0}^{π /2} \int_{0}^{R} r^{2} cos θ sin θ d r d θ = [\frac{r ^{3}}{3}]_{0}^{R} [\frac{s i n ^{2} θ}{2}]_{0}^{π /2} = \frac{R ^{3}}{6} .

The differential printed in the PDF is already

d r d θ

. No further polar Jacobian is to be inserted into this given iterated integral.

 New to topics? Read the docs here!