`y = sqrt(x - 1), y = 0, x = 5` Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. (about the x-axis)
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The volume of the solid obtained by rotating the region bounded by the curves `y=sqrt(x-1), y=0, x= 5` , about x axis, can be evaluated using the washer method, such that:
`V = int_a^b pi*(f^2(x) - g^2(x))dx`
Since the problem provides you the endpoint x = 5, you need...
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The parabola has vertex at (1,0) on the x axis.
So you are rotating that area enclosed by the parabola, the x-axis, and the vertical line x= 5.
Rotation about the x-axis gives the volume as : `int [ pi* R^2] dx` for x in [1,5]
If you rotate one point on the parabola, you should see a circle with radius = the height of the parabola.
So `R= sqrt(x-1)` and `R^2= (x-1) `
`int_1^5[ pi* (x-1)] dx `
`= pi [( x^2)/2 -x ] | _1^5`
`= pi[ (25/2 -5 ) -( 1/2 -1)] `
`= pi[ 12 -5 + 1] `
` = 8pi `
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