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Chapter 6 | Applications of Integration
Figure 6.35 (a) The region R bounded by two lines and the x -axis. (b) The solid of revolution generated by revolving R about the x -axis.
Looking at the region, if we want to integrate with respect to x , we would have to break the integral into two pieces, because we have different functions bounding the region over [0, 1] and [1, 2]. In this case, using the disk method, we would have V = ∫ 0 1 ⎛ ⎝ πx 2 ⎞ ⎠ dx + ∫ 1 2 ⎛ ⎝ π (2− x ) 2 ⎞ ⎠ dx .
If we used the shell method instead, we would use functions of y to represent the curves, producing
1
V = ∫
⎡ ⎣ ⎛ ⎝ 2− y ⎞
⎤ ⎦
⎛ ⎝ 2 πy
⎞ ⎠ dy
⎠ − y
0
1
= ∫
⎡ ⎣ 2−2 y ⎤ ⎦
⎛ ⎝ 2 πy
⎞ ⎠ dy .
0
Neither of these integrals is particularly onerous, but since the shell method requires only one integral, and the integrand requires less simplification, we should probably go with the shell method in this case. b. First, sketch the region and the solid of revolution as shown.
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