Question

Find the surface area of a sphere of radius r?

Answer 1

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 `r` by all coordinate points where `0 le phi le pi` (Where `phi` is the angle measured down from the positive `z`-axis), and `0 le theta le 2pi` (just the same as it would be polar coordinates), and `rho=r`).

The Jacobian for Spherical Coordinates is given by `J=rho^2 sin phi`

And so we can calculate the surface area of a sphere of radius `r` using a double integral:

`A = int int_R \ \ dS \ \ \`

where `R={(x,y,z) in RR^3 | x^2+y^2+z^2 = r^2 }`

`:. 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