7.3 volumes by cylindrical shells

Volumes by Cylindrical Shells

The volume of a right circular cylindrical shell with radius r, height h, and infinitesimal thickness dx, is given by:

Vshell = 2πrh dxIf one slits the cylinder down a side and unrolls it into a rectangle, the height of the rectangle is the height of the cylinder, h, and the length of the rectangle is the circumference of a circular end of the cylinder, 2πr. So the area of the rectangle (and the surface of the cylinder) is 2πrh. Multiply this by a (slight) thickness dx to get the volume.

In the diagram, the yellow region is revolved about the y-axis. Two of the shells are shown. For each value of x between 0 and a (in the graph), a cylindrical shell is obtained, with radius x and height f(x). Thus, the volume of one of these shells (with thickness dx) is given by Vshell = 2π x f(x) dx.

Summing up the volumes of all these infinitely thin shells, we get the total volume of the solid of revolution:

Example 1: Find the volume of the solid of revolution formed by rotating the region bounded by the x-axis and the graph of  from x=0 to x=1, about the y-axis.   

Example 2: Find the volume of the solid of revolution formed by rotating the finite region bounded by the graphs of and about the y-axis.

u = x – 1 so  x = u + 1du = dx

EXAMPLE  Find the volume of the solid obtained by rotating the region bounded by                  and             about the line           .

Time to Practice !!!

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