Rotate the triangle determined by the lines y = 5-x, x = 2, and the x-axis about the y-axis.

The figure is generated by rotating a triangle about the y-axis. In the left figure, the rectangles indicate the cylindrical shell shown on the right. Similar rectangles can be drawn over the whole figure, representing a series of concentric shells.

We can determine the volume of the figure by adding the volumes of the concentric shells. As indicated in the right figure the radius of each shell is the x-coordinate. Its height is the y-coordinate. The thickness of a shell is dx.

The volume, dV, of a shell is the (area of the curved surface) * thickness.

The area of the curved surface is the circumference * height. Hence,

. Substituting for y and integrating, we get

. If we multiply to get powers of x, we can use the Power Rule for integration:

. Now we apply the Fundamental Theorem of Calculus to get