Question

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. y=4x-x^2 y=3 about x=1

Answer 1

`(16\pi)/(3)`.

Explanation:

I graphed the region in the image below. The blue line is y=3, the purple line is x=1 and the green curve is y =
`4x-x^(2)`. The shaded region in blue is the region we are going to rotate.

Now, to find the volume v=
`2\pi \int\limits^a_b {p(x)h(x)} \, dx` where a=1, b=3 (left and right points of the region), p(x) is the distance from the rotation axis to the diferential Δx, we say p(x)=x and h(x) is the height of the region, in this case is h(x)=
`4x-x^(2)-3`. Then,

v =
`2\pi \int\limits^1_3 {x(4x-x^(2)-3)} \, dx`

=
`2\pi \int\limits^1_3 {4x^(2)-x^(3)-3x} \, dx`

=
`2\pi ((4x^(3))/(3)-(x^(4))/(4)-(3x^(2))/(2))^(3)_1`

=
`2\pi (36-(81)/(4)-(27)/(2)-(4)/(3)+(1)/(4)+(3)/(2))`

=
`2\pi*(8)/(3)`

=
`(16\pi)/(3)`.