Change to polar coordinates to evaluate: int(0 to 2) int(0 to sqrt(4-x^2)) sin x

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Explanation / Answer

y = sqrt(2x - x^2) ==> y^2 = 2x - x^2 ==> x^2 + y^2 = 2x (which may be rewritten as (x - 1)^2 + y^2 = 1, a unit circle with center (1, 0).) ==> r^2 = 2r cos ? ==> r = 2 cos ?. The upper half of this circle corresponds to ? in [0, p/2]. So, the integral equals ?(? in [0, p/2]) ?(r = 0 to 2 cos ?) sqrt(r^2) (r dr d?) = ?(? in [0, p/2]) ?(r = 0 to 2 cos ?) r^2 dr d? = ?(? in [0, p/2]) (8/3) cos^2(?) d? = ?(? in [0, p/2]) (4/3)(1 + cos(2?)) d? = (4/3)(? + sin(2?)/2) {for ? = 0 to p/2} = 2p/3.

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