Triple Integrals. Need help I cant figure out why 2 integrals arent matching up

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SUMMARY

The discussion centers on a student's confusion regarding the evaluation of a triple integral in Calculus 3, specifically Example 1 from their textbook. The student is attempting to reconcile their results with the expected outcomes, which involve integrating from 0 to π/2 versus 0 to π. The issue arises from the interpretation of the theta angle and the limits of integration, particularly with the radial component defined as r going from 0 to 2 sin(θ). This discrepancy is critical for achieving the correct final result.

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Triple Integrals
I am currently taking Calc 3 at college and I ran across a small problem with this one triple integral I am trying to work out (The problem is written as Example 1 in the book work document· So far I have had no luck in finding someone who could help me. If you can't help me it's okay, I am just trying to figure out what I am doing wrong. Attached is a copy of the problem from the book and how they did it with the final integration being from 0 to pi/2 and a copy of my work with the final integration being from 0 to pi. According to my professor it should be the same but I am not getting the same answer with this problem or the actual problem I have to do. Any help will be appreciated and if you can't help I understand.

Please try to help if you can...
 

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I think the theta angle is defined in a funny way in that exercise. If you have r going from zero to [tex]2 \sin \theta[/tex], then it goes from zero to zero.
 

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