Question

Find the volume generated by rotating the region bounded by y = x, x = 1, and y² = 4x, about the x-axis.

Answer 1

Given, the region is bounded by $y=x, y^2=4x$ and $x=1$ .

Volume generated is given by,

$v=\pi\int_a^b ([f(x)]^ 2−[g(x)] ^2)dx\\ =\pi\int_0^1((2\sqrt x)^ 2−x^ 2 )dx\\ =\pi\int_0^1(4x−x ^2)\\ =\pi(2x ^2− \frac{x^3}{3})_0^1 \\ =\pi(2− \frac{1}{3})\\ = \frac{5\pi}{3} ​$

Thus, volume is $\frac{5\pi}{3}$ .