Write a triple integral that gives the volume of the region between $z=2$ and the top half of the sphere $x^2 + y^2 + z^2 = 9 $

Question

The integral I've formed is:

$$\int_2^3 \int^{\sqrt{9-z^2}}_{-{\sqrt{9-z^2}}} \int^{{\sqrt{9-x^2-z^2}}}_{-{\sqrt{9-x^2 - z^2}}} dydxdz $$

Is this correct?

Even if it is, I'm not too sure why it would be correct. I've been thinking about these problems as first finding an integral for the area of the "shadow" that the region casts onto one of the planes (e.g. the $xy$-plane), and then adding a third integral to account for the last region. However, in this question, the inner most integral uses two variables in its limits...

It is correct. Your interpretation of what an integral does is a little flawed. You can only find the area of a shadow and then just add a third integral if the bounds don't bend in that direction (like how a straight up and down cylinder doesn't "bend" in the z direction). In this case, in order to capture the volume the innermost bounds must go from a surface to a surface, which should be a function of two variables. — Ninad Munshi, Sep 24 '19 at 15:59
do you mean the volume between $z=2$ and the top of the sphere (it's not half)? cf. this recent popular related question — J. W. Tanner, Sep 24 '19 at 16:02

Write a triple integral that gives the volume of the region between $z=2$ and the top half of the sphere $x^2 + y^2 + z^2 = 9 $

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