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$$\int_{-1}^{1}\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\sqrt{4-x^2-y^2} \, dy \, dx$$

Is there a way to calculate this definite integral by hand?

Michael Hardy
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greg lee
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    Polar coordinates might help. – Julian Rosen Sep 08 '13 at 02:55
  • For this type of question it is best to start by drawing a picture. Once you've done this, it becomes obvious that this problem was not meant to be done in rectangular coordinates. – Ryan Sep 08 '13 at 02:59

1 Answers1

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Drawing the region, we see that it's a disk of radius $1$ around the origin. Make a change of variables into polar coordinates, getting

$$\int \int_{D(0, 1)} \sqrt{4 - x^2 - y^2} dy dx = \int_0^{2\pi} \int_0^1 \sqrt{4 - r^2} r dr d\theta = 2\pi \int_0^1 r \sqrt{4 - r^2}dr$$

Now an easy substitution allows the last integral to be evaluated.