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

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- Solution 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.

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