Question

If a region is bounded by `y = sqrt(x) + 3`, `y=5`, and the y=axis and it is revolved around the y = 7, how do you find the volume?

Answer 1

The volume is `=120pi`

Explanation:

The volume of a small slice,

`dV=pi(7-(sqrtx+3))^2dx`

Therefore,

`V=int_0^4pi(7-(sqrtx+3))^2dx`

`V=piint_0^4 (49-(x+6sqrtx+9))dx`

`=piint_0^4 (49-x-6sqrtx-9)dx`

`=pi[40x-x^2/2-6x^(3/2)/(3/2)]_0^4`

`=pi(160-8-32)=120pi`

graph{(y-(sqrtx+3))(y-5)(y-7)=0 [-5.335, 10.465, 2.226, 10.126]}