Definite Integration: Volume

pp123123

2012-01-31 2:50 am

Disk Method

If the region bounded by the parabola y=4-(x+5)^2 and the x-axis is revolved about the y-axis, find the volume of the solid of revolution.

In this case, as the solid is revolved about the y-axis, should

V=Int pi (x1^2-x2^2) [d,c] be used? but there is no x2?

更新1:

sorry should beV=Int pi (x1^2-x2^2)dy [d,c]

回答 (1)

用戶10012

2012-01-31 3:30 pm

✔ 最佳答案

You have to cut the curve into 2 parts along the axis of symmetry, one part is x2 and the other part is x2.
y = 4 - (x + 5)^2
(x + 5)^2 = 4 - y
x + 5 = +/- sqrt ( 4 - y)
x = - 5 + sqrt (4 - y) and x = - 5 - sqrt (4 - y)
- 5 + sqrt(4 - y) is x2 because it is closer to the y - axis. - 5 - sqrt(4 - y) is x1 because it is farther from the y - axis.

2012-01-31 07:32:10 補充：
Correction : 2nd line should be " and the other part is x1".

2012-01-31 07:34:30 補充：
Remark : Shell method could be simpler when it is not easy to cut the curve.

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