A spherical planet of radius \(a\) has a variable density \(f(r)\) which depends only on the distance \(r\) from the planet’s centre. Show that the average density of the planet is

\[3 \int_0^1 t^2 f(at) \:dt.\]

The tricky part is going to be finding the total mass of the planet. Since the density depends only on \(r\) we should think about the mass of a thin hollow spherical shell.

Find the average density correct to two significant figures in each of the three cases:

- \(f(r) = \exp\left[\left(-\dfrac{r}{a}\right)^3\right]\), where \(\exp(x)\) denotes \(e^x\),

We will need to use the result from the first part.

- \(f(r) = \exp\left(-\dfrac{r}{a}\right)\),

We will need to find a tool to help us evaluate the integral we get this time.

- \(f(r) = \dfrac{a^2r}{(a+r)^3}\).

This time the integral involves an awkward looking fraction. Again, we’ll need to find an appropriate tool to help.