Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

y = 3 − 3x^2, y = 0

Answer 1

Answer:
`9.6\pi`

Explanation:

Given

Curve is
`y=3-3x^2`

boundary is y=0 i.e.

`\Rightarrow 0=3-3x^2\\\Rightarrow x^2=1\\\Rightarrow x=\pm 1`

The volume of solid generated when rotated about the x-axis is

`\Rightarrow V=\int_a^b\pi y^2dx`

Putting values we get

`\Rightarrow V=\int_(-1)^(1)\pi (3-3x^2)^2dx\\\\\Rightarrow V=\int_(-1)^(1)\pi(9+9x^4-18x^2)dx\\\\\Rightarrow V=\pi \left [ (9x^5)/(5) - 6 x^3 + 9 x\right ]_(-1)^(1)\\\\\Rightarrow V=9.6\pi`