Volumes by Integration: Shell Method Practice Problem Solution

3. Take the area bounded by the x axis and y=5x-x^{2}-4 and rotate it about the y axis and determine the volume.

Here is the volume. It’s sort of a half-bagel shape.

Then write the equation for dV=2pr*height*thickness

dV=2p(5x-x^{2}-4)xdx

Now set limits and integrate. To set the limits, we have to determine where the curve intersects the x axis first.