Review question

# What is the average density of this spherical planet? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8152

## Question

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