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How to evaluate Surface Integral?
Evaluate the surface integral.
∫∫S (x^2z + y^2z) dS
S is the hemisphere x^2 + y^2 + z^2 = 1, z ≥ 0.
1 Answer
- Ivan ALv 61 decade agoFavorite Answer
The integrand can be rewritten as
(x^2z + y^2z) = z (x^2 + y^2) = z (1-z^2)
where I used the fact that the integral is over the surface ^2 + y^2 + z^2 = 1, which imposes a constraint.
dS can be rewritten as
dS = 2pi (1-z^2) (1) d theta = 2pi (1-z^2) (1) dz / sqrt(1-z^2)
dS = 2pi sqrt(1-z^2) dz
pay special attention to how I set up the the surface differential; you will need t o make a sketch to figure this one out. So the surface integral turns into a simple linear integral given by
integral(0,1) [ 2pi z (1-z^2)^(3/2) dz ]
which easily integrates. You should figure out the rest.
