Finding volume of a sphere

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?

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?