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We have a sphere with the center coordinates $(0,0,0)$ and a radius $a$. If we separate this sphere with the plane $y= a/2$ what will be the volume of the region of the sphere between $y=a/2$ and $y=a$. I tried to use cylindrical coordinates but I think I could not succeed at that.

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    have a look at (https://en.wikipedia.org/wiki/Spherical_cap) – 2017-01-13

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This is a solid of revolution with circular cross-sections. It will just be

$$ \int_{a/2}^a\pi x^2\,dy $$

with $x^2=a^2-y^2$.

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    How can we write this with triple integrals? – 2017-01-13
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    The outer integral with respect to $y$ will be the same but the $\pi x^2$ will have to be set up as a double integral. You can do just one fourth of the circle and multiply by $4$ because of the symmetry. I assume that you can find the area of a circle using a double integral? – 2017-01-13