Chapter 6 | Applications of Integration
657
Figure 6.26 (a) A representative rectangle. (b) When this rectangle is revolved around the y -axis, the result is a cylindrical shell. (c) When we put all the shells together, we get an approximation of the original solid.
To calculate the volume of this shell, consider Figure 6.27 .
Figure 6.27 Calculating the volume of the shell.
The shell is a cylinder, so its volume is the cross-sectional area multiplied by the height of the cylinder. The cross-sections are annuli (ring-shaped regions—essentially, circles with a hole in the center), with outer radius x i and inner radius x i −1 . Thus, the cross-sectional area is πx i 2 − πx i −1 2 . The height of the cylinder is f ( x i * ). Then the volume of the shell is V shell = f ( x i * )( πx i 2 − πx i −1 2 ) = πf ( x i * ) ⎛ ⎝ x i 2 − x i −1 2 ⎞ ⎠ = πf ( x i * )( x i + x i −1 )( x i − x i −1 ) =2 πf ( x i * ) ⎞ ⎠ ( x i − x i −1 ). ⎛ ⎝
x i + x i −1 2
Note that x i − x i −1 =Δ x , so we have
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