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Chapter 6 | Applications of Integration
Figure 6.20 (a) Shown is a thin rectangle between the curve of the function g ( y ) = 4− y and the y -axis. (b) The rectangle forms a representative disk after revolution around the y -axis.
The region to be revolved and the full solid of revolution are depicted in the following figure.
Figure 6.21 (a) The region to the left of the function g ( y ) = 4− y over the y -axis interval [0, 4]. (b) The solid of revolution formed by revolving the region about the y -axis.
To find the volume, we integrate with respect to y . We obtain V = ∫ c d π ⎡ ⎣ g ( y ) ⎤ ⎦ 2 dy = ∫ 0 4 π ⎡ ⎣ 4− y ⎤ ⎦ 2 dy = π ∫ 0 4 ⎛ ⎝ 4− y ⎞ ⎠ dy = π =8 π .
⎤ ⎦ ⎥ |
⎡ ⎣ ⎢ 4 y −
0 4
y 2 2
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