Question

The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. y = −x2 + 17x − 72, y = 0; about the y−axis

Answer 1

The volume is 17π/6

Explanation:

We are going to apply first the shell method where the slice is parallel to the axis of revolution, then we need to determine the medium radius ahd the height of the shell, in this case as the region is rotated about the y-axis
`R=x` and the height is
`h=-x^(2) +17x-72`

Then the integral is
`V=2\pi \int\limits^a_b {radius*height} \, dx`

`V=2\pi \int\limits^8_9 {x*(-x^(2)+17x-72)} \, dx`

`V=2\pi [-(x^(4) )/(4) +(17)/(3)x^(3)-36x^(2) ]_8^9`

`V=2\pi [-(2465)/(4)+(3689)/(3)-612 ]`

`V=(17\pi )/(6)u^(3)`

To show the answer we can apply the ring method where the the inner radius is
`r=(17)/(2)-17\sqrt{(1)/(4)-y }` and the outter radius
`R=(17)/(2)+\sqrt{(1)/(4)-y }`

The formula is
`V=\pi \int\limits34 \sqrt{(1)/(4)-y }dy` evaluated between
`y=0` and
`y=(1)/(4)`

Solving the integral we can get the same solution of
`(17\pi )/(6) u^(3)`

Finally, the volumen is
`(17\pi )/(6) u^(3)`