Sketch the figure.
If you slice the region vertically, the height of the shell doesn’t change because of the shift in axis of
revolution, but you have to add 1 to each radius.
Our integral thus becomes
2 π (x + 1)( ) dxThe last formula you need to learn involves slicing the region horizontally and revolving it about the x-
axis. As you probably guessed, you’ll get a cylindrical shell.
If you have a region whose area is bounded on the right by the curve x = f(y) and on the left by the curve x
= g(y), on the interval [c, d], then each cylinder will have a height of f(y) − g(y), a radius of y, and an area
of 2πy[f(y) − g(y)].
To find the volume, evaluate the integral.
2 π y[(f(y) − g(y)] dyThis is the formula for finding the volume using cylindrical shells when the region is rotated around the x-
axis.
Example 7: Find the volume of the region that results when the region bounded by the curve x = y^3 and
the line x = y, from y = 0 to y = 1, is rotated about the x-axis. Set up but do not evaluate the integral.