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Find the volume of the solid generated by revolving the region bounded by the given...

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dylachan | Student | (Level 1) Honors

Posted December 4, 2011 at 1:57 PM via web

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Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.

y=6cos(pi x), y=0, x=-0.5, x=0.5

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted December 4, 2011 at 4:11 PM (Answer #1)

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Remember the formula of the volume of solid of revolution:

V = pi*`int_a^b y^2 dx`

`y = 6 cos(pi x) =gt y^2 = 36 cos^2(pi x)`

V = pi*`int_-0.5^(0.5) 36 cos^2(pi x)dx` = 2pi*`int_0^(0.5)36 cos^2(pi x)dx` ( `cos^2 pi x`  is an even function)

Use `cos^2 alpha = (1 + cos 2alpha)/2 =gt cos^2(pi x) = (1 + cos 2pi*x)/2`

V = 2pi * 36*`int_0^(0.5)(1+cos 2pi x)/2dx`

V = 2pi*36*(`int_0^(0.5)(1/2)dx`  + `int_0^(0.5) cos ((2pix)/2)dx` )

`V = 2pi*36*(1/2)*( 0.5 - 0 + (sin 2pi*0.5)/(2pi) - (sin 0)/(2pi))`

sin 0 = 0

`V = 36 pi* (1/2 + (sin 2pi*(1/2))/(2pi))`

`sin pi = 0`

V = `36pi/2 =gt V = 18pi`

The volume of the solid of revolution is V = `18pi.`

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