Spherical Integration Practice

Calculus Level 3

Compute the integral of the function arctan ⁡ ( y x ) \arctan \left (\frac yx \right) over the volume contained inside x 2 + y 2 + z 2 = 9 x^2+y^2+z^2 = 9 .

1 1 3 π 2 3\pi^2 12 π 12\pi 36 π 2 36\pi^2

Matt DeCross by Matt DeCross , 27, USA

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1 solution

Matt DeCross
Apr 25, 2016

The integral is best done in spherical coordinates:

∫ 0 2 π ∫ 0 π ∫ 0 3 θ r 2 sin ⁡ ϕ d r d ϕ d θ = 9 π ∗ 2 ( ( 2 π ) 2 2 ) = 36 π 2 . \int_0^{2\pi} \int_0^{\pi} \int_0^3 \theta r^2 \sin \phi dr d\phi d\theta= 9\pi * 2 \left(\frac{(2\pi)^2}{2}\right) = 36 \pi^2.

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