Find the surface area of a sphere of radius r?

Question

Cameron Alexander · Accepted 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 #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

Elijah Abram · Answer

undefined The surface area $A$ of a sphere with radius $r$ is given by the formula:

$$ A = 4 \pi r^2 $$