I don't understand why theta and thy range from 0 to pi/2. shouldn't it be from 0 to pi/4 since you divide the whole sphere by 8? a full circle is 2pi, so eight of a circle should be pi/4. Down below is the question and the answer key
2026-04-01 20:25:17.1775075117
What would be the bound of integration for this triple integral?
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You're not taking an eighth of the range of these angles—you're taking an eighth of the the sphere's volume. In order to get this, you need to take a quarter of the angle range for $\theta$ and half of the range for $\phi$.
In order to wrap your head around this, keep in mind the visual intuition behind $\theta$ and $\phi$ with spherical coordinates:
You are right, to a degree. The full range of $\theta$ is $0\leq\theta<2\pi$, which visually represents the unit circle on the xy-plane; the full range for $\phi$ is $0\leq\phi\leq\pi$, which visually represents half of the unit circle on the yz-plane.
Notice that if $0\leq\theta\leq\frac{\pi}{2}$, you restrict the range of the sphere to the first quadrant of the xy-plane, which is a quarter of the entire xyz-plane. If you restrict $0\leq\phi\leq\frac{\pi}{2}$, you get only positive values for $z$, giving you an eighth of the xyz-plane.