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

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

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

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

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