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Chapter 6 | Applications of Integration
V shell =2 πf ( x i * ) ⎛ ⎝
⎞ ⎠ Δ x .
x i + x i −1 2
x i + x i −1 2
is both the midpoint of the interval [ x i −1 , x i ] and the average radius of the shell, and we can
Furthermore,
approximate this by x i * . We then have
V shell ≈2 πf ( x i * ) x i * Δ x . Another way to think of this is to think of making a vertical cut in the shell and then opening it up to form a flat plate ( Figure 6.28 ).
Figure 6.28 (a) Make a vertical cut in a representative shell. (b) Open the shell up to form a flat plate.
In reality, the outer radius of the shell is greater than the inner radius, and hence the back edge of the plate would be slightly longer than the front edge of the plate. However, we can approximate the flattened shell by a flat plate of height f ( x i * ), width 2 πx i * , and thickness Δ x ( Figure 6.28 ). The volume of the shell, then, is approximately the volume of the flat plate. Multiplying the height, width, and depth of the plate, we get V shell ≈ f ( x i * ) ⎛ ⎝ 2 πx i * ⎞ ⎠ Δ x , which is the same formula we had before. To calculate the volume of the entire solid, we then add the volumes of all the shells and obtain V ≈ ∑ i =1 n ⎛ ⎝ 2 πx i * f ( x i * )Δ x ⎞ ⎠ . Here we have another Riemann sum, this time for the function 2 πxf ( x ). Taking the limit as n →∞ gives us
→∞ ∑ i =1 n
b
⎠ = ∫
⎛ ⎝ 2 πx i * f ( x i * )Δ x ⎞
⎛ ⎝ 2 πxf ( x ) ⎞
V = lim n
⎠ dx .
a
This leads to the following rule for the method of cylindrical shells.
Rule: The Method of Cylindrical Shells Let f ( x ) be continuous and nonnegative. Define R as the region bounded above by the graph of f ( x ), below by the x -axis, on the left by the line x = a , and on the right by the line x = b . Then the volume of the solid of revolution
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