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Sleepycoaster
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Homework Statement
I know how to find the volume of a sphere just by adding the areas of circles, so I decided to do a double integral to find the same volume, just for fun.
Here's what I've set up. I put 8 out front and designed the integrals to find an eighth of a sphere that has its center at the origin. This piece of the sphere is what you would get if you cut a sphere in half three times, one from each of the three dimensions.
8∫01∫0√(1-y2)√(1-x2-y2)dxdy
√(1-x2-y2) is the equation I used to measure the z-dimension of a point on the sphere given values for x and y.
Homework Equations
The Attempt at a Solution
Here's the integral I got for the inner integration, the one with respect to x:
∫0√(1-y2)√(1-x2-y2)dx
=[-2x/(2√(1-x2-y2))] with 0 on the bottom and √(1-y2) on the top.
As you can see, if I plug in √(1-y2) for x, the values cancel out so that there is only a zero in the denominator.
Is this simply a limitation in using multiple integration, or did I do something wrong? Any help is appreciated.
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