 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-v-resit/question-1/a/ii/solution

\overline{r} = \overline{x} i + \overline{y} j

. The region has reflection symmetry in the

y

-axis: replacing

θ

π - θ

changes

x

- x

and preserves

y

and

d A

\iint_{R} x d A = 0, \overline{x} = 0.

y = r sin θ

\iint_{R} y d A = \int_{0}^{π} \int_{0}^{s i n^{2} θ} r^{2} sin θ d r d θ = \frac{1}{3} \int_{0}^{π} sin^{7} θ d θ = \frac{32}{105} .

Here the last integral has the value allowed in the paper. Dividing the first moment of area by the area found in i gives

(\overline{x}, \overline{y}) = (0, \frac{512}{315 π}), \overline{r} = \frac{512}{315 π} j .

The positive vertical coordinate agrees with the fact that the entire region lies above the

x

-axis.

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