Substituting spherical coordinates to evaluate an integral

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The integral in question involves evaluating a three-dimensional volume using spherical coordinates. The initial attempt yielded an integral that evaluates to 3.57, while the original Cartesian integral evaluates to 5.236. There is a discussion about the geometric interpretation of the volume, which includes a sphere for z<0 and a cylinder for z>0. It is suggested that a combination of spherical and cylindrical coordinates might be necessary to accurately represent the volume. The conclusion emphasizes the complexity of the problem and the need for careful consideration of the geometry involved.
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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.
 
Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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