Find the surface integral?

Find the surface integral ∫ ∫ f dS of the function f(x,y) = x^2 + y^2 over the sphere x^2 + y^2 + z^2 = 1.

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  • 8 years ago
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    ∫ ∫ f dS

    = ∫ ∫ f(r(u,v)) || ∂r/∂u x ∂r/∂v || dA

    r(u,v) = {cos(u) sin(v), sin(u) sin(v), cos(v)} ... is the unit sphere

    || ∂r/∂u x ∂r/∂v || = sin(v)

    = ∫ ∫ sin³(v) du dv

    {(u,v) | 0 ≤ u ≤ 2π ; 0 ≤ v ≤ π}

    = 2π ∫ sin³(v) dv

    {(v) | 0 ≤ v ≤ π}

    = 8π/3

    Answer: 8π/3

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