Surface integral of half sphere

[JamesMay]

Homework Statement

I am trying to sort out surface integrals in my head, and have become more confused when attempting to calculate the surface integral of a hemisphere. I am getting confused about which values to use as boundaries.

Homework Equations

da=R^2 sinθdθdφ

The Attempt at a Solution

To get half a sphere I integrated dφ between 0 and 2pi, and integrated sinθdθ between 0 and pi/2.

This gives R^2*0*2pi=0

Whereas if I integrate dφ between 0 and pi, and integrate sinθdθ between 0 and pi i get

R^2*1*pi

which is the answer I am looking for but shouldn't I get the same using both sets of boundary values? Confused...

 

[HallsofIvy]

? Not if you integrate correctly! d\phi, integrated from 0 to 2\pi is, of course, 2\pi and sin\theta d\theta integrated from 0 to \pi/2 is -cos\theta evaluated between 0 and 2\pi: -(0- (1))= 1, not 0. The half sphere area is 2\pi R^2.

If, instead, you integrate d\phi from 0 to \pi and sin\theta d\theta from 0 to \pi you get \pi for the first integral and -cos(\theta) evaluated from 0 to \pi, which is -(-1- (1))= 2 for the second, again getting 2\pi R^2.

 

[JamesMay]

Oh right, I always forget that cos(0) isn't just 0. You would think I'd have learned that by now. Thanks!