 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/9/solution

In spherical coordinates, integrating over the angular variables gives the volume element

4 π r^{2} d r

for a spherical shell. The mass density therefore gives the total mass

M = 4 π \int_{0}^{a} (1 + r) r^{2} d r = 4 π [\frac{r ^{3}}{3} + \frac{r ^{4}}{4}]_{0}^{a} .

Thus

M = \frac{4 π a ^{3}}{3} + π a^{4} .

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