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Chapter 6 | Applications of Integration
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 formed by revolving R around the x -axis is given by (6.3) V = ∫ a b π ⎡ ⎣ f ( x ) ⎤ ⎦ 2 dx .
The volume of the solid we have been studying ( Figure 6.18 ) is given by V = ∫ a b π ⎡ ⎣ f ( x ) ⎤ ⎦ 2 dx = ∫ −1 3 π ⎡ ⎣ ( x −1) 2 +1 ⎤ ⎦ 2 dx = π ∫ −1
3 ⎡ ⎣ ( x −1) 4 +2( x −1) 2 +1 ⎤ ⎦ dx
⎦ | −1 3
⎡ ⎣ 1 5
( x −1) 3 + x ⎤
⎡ ⎣
− 16 3 −1 ⎞ ⎠ ⎤
⎛ ⎝ 32 5
+ 16 3 +3 ⎞
⎛ ⎝ − 32 5
⎦ = 412 π 15
( x −1) 5 + 2 3
units 3 .
⎠ −
= π
= π
Let’s look at some examples.
Example 6.8 Using the Disk Method to Find the Volume of a Solid of Revolution 1
Use the disk method to find the volume of the solid of revolution generated by rotating the region between the graph of f ( x ) = x and the x -axis over the interval [1, 4] around the x -axis.
Solution The graphs of the function and the solid of revolution are shown in the following figure.
Figure 6.19 (a) The function f ( x ) = x over the interval [1, 4]. (b) The solid of revolution obtained by revolving the region under the graph of f ( x ) about the x -axis.
We have
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