Question

Consider the region bounded by the curves 2. y = 3 + x , y = 3, and x = 2. Find the volume of the solid obtained by rotating this region about the axis y = 3.

Answer 1

`(8\pi)/(3)` units cubed

Explanation:

Let's look at a point (x,y) on the line y=3+x. The height between (x,y) and the x-axis is y. We want the distance from the axis of rotation which is y=3 so the height (or distance between) point (x,y) on y=3+x and y=3 is y-3.

y-3 is the radius in terms of y.

(3+x)-3=x is the radius in terms of x. I replaced y with 3+x since we have y=3+x.

The area of the circle I drew is
`\pi \cdot r^2=\pi \cdot x^2`

To find the volume we must integrate the area of the circle we found between the bounded lines x=0 and x=2.

`\int_0^2 \pi \cdot x^2`

`\pi \cdot (x^3)/(3)|_0^2`

`\pi[(2^3)/(3)-(0^3)/(3)]`

`\pi[(8)/(3)-0]`

`\pi[(8)/(3)]`

`(8\pi)/(3)` units cubed