(Vector Calculus) Help regarding area element notation

[alqubaisi]

Homework Statement

The area element of a sphere in spherical coordinates is given as following

$dA = r^2 \sin(\phi)\; d \theta \; d \phi$​

using the notation in the following figure:

However, while going through some E&M books I ran into the following notation

$Surface \; Area = r^2 \; \int_{-1} ^1 d \cos(\phi) \; \int_0^{2 \pi}d \theta \; = 4 \pi r^2$​

My question is how can we replace $\int_{0} ^\pi \sin(\phi) \; d \phi$ with $\int_{-1} ^1 d \cos(\phi)$

 

[HallsofIvy]

That's basically a substitution. If you let $u= cos(\phi)$ then $du= d(cos(\phi)= -sin(\phi)d\phi$. Also, when $\phi= 0$, $cos(\phi= 1$ and when $\phi= \pi$, $cos(\phi)= -1$.

With that substitution, $\int_0^\pi sin(\phi)d\phi= \int_1^{-1} -du$ and, of course, swapping the limits of integration multiplies the integral by -1:
$\int_0^\pi sin(\phi)d\phi= \int_1^{-1} -du= \int_{-1}^1 du= \int_{-1}^1 d(cos(\phi))$