Volumes by Integration: Shell Method Practice Problem Solution

2. Rotate the area bounded by the x and y axes and y=3-x^{2} about the y axis and find the volume.

Here is the volume with the shell sketched in:

Now write the equation for the volume of the shell, dV:

dV=2pr*height*thickness

dV=2py(4-x)dy

Now substitute for x in terms of y, set limits, and integrate.