Substituting spherical coordinates to evaluate an integral

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SUMMARY

The discussion focuses on evaluating the integral $$\int^1_{-1} \int^{ \sqrt {1-x^2}}_{ - \sqrt {1-x^2}} \int^1_{-\sqrt{x^2+y^2}}dzdydx$$ using spherical coordinates. The participant transformed the integral into spherical coordinates resulting in $$\int^1_{0} \int^{ 2\pi}_0 \int^{3\pi/4}_{0}r^2\sin\theta d\phi d\theta dr$$, yielding a value of 3.57. However, the original integral evaluates to 5.236, indicating a discrepancy. The discussion raises questions about the appropriateness of using spherical coordinates for the lower half of the volume and cylindrical coordinates for the upper half.

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I have to evaluate

$$\int^1_{-1} \int^{ \sqrt {1-x^2}}_{ - \sqrt {1-x^2}} \int^1_{-\sqrt{x^2+y^2}}dzdydx$$

using spherical coordinates.

This is what I have come up with

$$\int^1_{0} \int^{ 2\pi}_0 \int^{3\pi/4}_{0}r^2\sin\theta d\phi d\theta dr$$

by a combination of sketching and substituting spherical coordinates.

After evaluating I obtain this integral to equal 3.57.

where as the first one evaluates to 5.236.

These are so difficult :(
 
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I am not convinced you did that right.
The volume of integration appears to be a sphere for z<0 and a cylinder for z>0. Your spherical integral doesn't look like that.

Are you allowed to use spherical integral for the lower half and cylindrical integral for the upper half? Or maybe just geometry...1/2 volume of unit sphere + volume of unit cylinder = 5.236.
 

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