# Find the surface area of a sphere of radius r?

##### 1 Answer

It is easier to use Spherical Coordinates, rather than Cylindrical or rectangular coordinates. This solution looks long because I have broken down every step, but it can be computed in just a few lines of calculation

With spherical coordinates, we can define a sphere of radius

The Jacobian for Spherical Coordinates is given by

And so we can calculate the surface area of a sphere of radius

# A = int int_R \ \ dS \ \ \ #

where

# :. A = int_0^pi \ int_0^(2pi) \ r^2 sin phi \ d theta \ d phi#

If we look at the inner integral we have:

# int_0^(2pi) \ r^2 sin phi \ d theta = r^2sin phi \ int_0^(2pi) \ d theta #

# " " = r^2sin phi [ \ theta \ ]_0^(2pi)#

# " " = (r^2sin phi) (2pi-0)#

# " " = 2pir^2 sin phi#

So our integral becomes:

# A = int_0^pi \ 2pir^2 sin phi \ d phi#

# \ \ \ = -2pir^2 { cos phi ]_0^pi#

# \ \ \ = -2pir^2 (cospi-cos0)#

# \ \ \ = -2pir^2 (-1-1)#

# \ \ \ = -2pir^2 (-2)#

# \ \ \ = 4pir^2 \ \ \ # QED