Volumes by Integration: Disk Method

Now sketch a differential element, a volume disk, with thickness dx.

OK. Let’s write the equation for the volume of the disk, dV:

dV=pr^{2}dx

Now we have to get the whole equation in terms of x. Note that the radius of the disk, r = y. And y = 2x^{1/2}. Let’s plug this into the equation for dV.

dV=p(2x^{1/2})^{2}dx = p4xdx

Now we can integrate. This time, we’ll include limits. Note that the disk would sweep along the x axis from 0 to 4.