I am stuck on the following problem:
A circle $x^2 + y^2 = a^2$ is rotated around the y-axis to form a solid sphere of radius a. A plane perpendicular to the y-axis at $y = \frac {a}{2}$ cuts off a spherical cap from the sphere. What fraction of the total volume of the sphere is contained in the cap?
So far I have figured out the following:
Rotating the cap on the y axis we get a height $h$ starting from $y = \frac {a}{2}$. The interval from $y = 0$ to $y = \frac {a}{2}$ (the region below the cap) should be:
$a - h$
I also know that the radius of the sliced disk, $x$, can be derived from the equation of the circle: $x = \sqrt {a^2 - y^2}$
Since the area of a circle is $A = \pi r^2$ the area with respect to $y$ for the circle should be:
$A(y) = \pi (a^2 - y^2)$
So to find the volume, we need to integrate the function:
$V = \int_\frac{a}{2}^a \pi (a^2 - y^2) dy$
I know where I should go, but I am not sure what to do about the constraint $y = \frac {a}{2}$ at this point. Should I integrate the terms with respect to y first and then plug in the value which is equal to y? Or should this be done before integrating?
Thanks.