(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))