The triangle determined by the lines y =x, y = 2, and x = 5 is rotated about the x-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 y-coordinate of the slanted line in the left diagram. Its length is the difference between the right and left edges. XR = 5 for all shells, and XL = y (from the equation of the slanted line). The thickness of a shell is dy.

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

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

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

. Now we apply the Fundamental Theorem of Calculus to get