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7.3 volumes by cylindrical shells
- 1. Volumes by CylindricalShells
- 2. The volume ofa right circular cylindrical shell with radius r, height h, and infinitesimal thickness dx, is given by: Vshell = 2πrh dx If 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.
- 3. 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.
- 4. Summing up thevolumes of all these infinitely thin shells, we get the total volume of the solid of revolution: V = 2pxf (x)dx = 2p xf (x) 0 a ò 0 a ò dx
- 5. Example 1: Findthe 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. y = x V = 2px x dx = 2p x 3 2 0 1 ò 1 2 ò dx = 2p × 2x 5 2 5 ù û ú ú ú0 1 = 4p 5 1 5 2 - 0 5 2 æ è ç ö ø ÷ ù û ú ú0 1 = 4p 5
- 6. Example 2: Findthe volume of the solid of revolution formed by rotating the finite region bounded by the graphs of and about the y-axis. y = x -1 y = x -1 ( ) 2
- 7. V = 2pxx -1- x -1 ( ) 2 ( )dx = 1 2 ò 2p x x -1dx - x x -1 ( ) 2 dx 1 2 ò 1 2 ò æ è ç ö ø ÷= 2p u+1 ( ) u du- u+1 ( )u2 du ò ò æ è ç ö ø ÷ = u = x – 1 so x = u + 1 du = dx 2p u 3 2 +u 1 2 æ è ç ö ø ÷du- u3 +u2 ( )du ò ò æ è ç ç ö ø ÷ ÷ = 2p 2u 5 2 5 + 2u 3 2 3 - u4 4 - u3 3 é ë ê ê ê ù û ú ú ú = 2p 2 x -1 ( ) 5 2 5 + 2 x -1 ( ) 3 2 3 - x -1 ( ) 4 4 - x -1 ( ) 3 3 é ë ê ê ê ù û ú ú ú1 2 = 2p 2 5 + 2 3 - 1 4 - 1 3 é ë ê ù û ú= 2p 29 60 = 29p 30
- 8. EXAMPLE Find thevolume of the solid obtained by rotating the region bounded by and about the line . Time to Practice !!! y = x - x2 y = 0 x = 2
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