Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 9

Table of contents
- Solution 9

Solution

For a spherically symmetric mass density, the angular integral in spherical coordinates is

4 π

. The two disjoint shell contributions therefore give the mass

M = 4 π \int_{a}^{b} r^{2} d r + 4 π \int_{c}^{d} r^{- 2} d r,

so

M = 4 π [\frac{b ^{3} - a ^{3}}{3} + \frac{1}{c} - \frac{1}{d}] .

Both contributions are positive because

a < b

and

c < d

.

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