Area again

Calculus Level 4

Find the area of the portion of the cylinder x 2 + y 2 = 6 y x^2+y^2=6y lying inside the sphere x 2 + y 2 + z 2 = 36 x^2+y^2+z^2=36 .


The answer is 144.

Jose Sacramento by Jose Sacramento , 66, Portugal

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1 solution

Otto Bretscher
Mar 7, 2016

We can parameterize the circle C C given by x 2 + y 2 = 6 y x^2+y^2=6y as x = 3 cos ⁡ t , y = 3 + 3 sin ⁡ t x=3\cos t,y=3+3\sin t and observe that for the sphere we have z = 36 − x 2 − y 2 = 6 sin ⁡ ( t 2 ) z=\sqrt{36-x^2-y^2}=6\sin(\frac{t}{2}) . Now the area we seek is the line integral 2 ∫ C 36 − x 2 − y 2 d s = 2 × 3 ∫ 0 2 π 6 sin ⁡ ( t 2 ) d t = 144 2\int_{C}\sqrt{36-x^2-y^2}ds=2\times3\int_{0}^{2\pi}6\sin(\frac{t}{2})dt=\boxed{144}

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Moderator note:

Great solution! Parametrizing the circle and then accounting for the volume element makes it easier to understand what is happening.

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