PracticeTest3 #2

Hello professor,
I was working on the Exam 3 practice ( https://math.asu.edu/sites/default/files/mat_267-post_exam3_solutions_0.pdf ) and I was confused on question #2 of the multiple choice. For the limits of integration of phi I got found them to be from 0 to pi. However, the correct answer says its from 0 to pi/2. Why is that?

Thank you

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In Cartesian (xyz) coordinates, your integration dz integration runs from z=0 to z=√(4-x^2-y^2); i.e. from the xy-plane UP to the top half of the sphere x^2+y^2+z^2=4, specifically there's nothing below the xy-plane. But the xy-plane is at 90 degrees or π/2 radians from the positive z-axis, hence π/2 has to be the upper limit.  If you went all the way to π your domain would have to include the negative z axis.