Question

Find the volume of the solid formed by rotating the region bounded by the graph of y equals 1 plus the square root of x, the y-axis, and the line y = 3 about the line y = 5.

Answer 1

Volume of solid is
`(40)/(3)\pi`

Explanation:

We need to find the volume of solid formed by rotating the region bounded by the graph of
`y=1+sqrt{x}`, y-axis and the line y=3 about the line y=5.

Please see the attachment for figure.

Using Shell method,
`V=\int_a^b\pi (R^2-r^2)dx`

where,

a=0 (Lower limit of solid)

b=4 (Upper limit of solid)

`R=4-√(x)` (Outer Radius of Shell)

r=2 (Inner radius of shell)

dx is thickness of shell

Volume of shell, dV=Area of shell x Thickness

Volume of solid
`V=\int dV`

`V=\int_0^4 \pi [(4-√(x))^2-2^2]dx`

`V=\int_0^4 \pi (16+x-8√(x)-4)dx`

`V=\pi (12x+(x^2)/(2)-(16)/(3)x^(3/2)|_0^4`

`V=\pi (48+8-(128)/(3))`

`V=(40)/(3)\pi`

Thus, Volume of solid is
`(40)/(3)\pi`