Question

Find the volume generated if the region enclosed by y = x² and the line y = 2x is revolve about the x-axis. Answer in 2 decimal places.

Answer 1

$y=x^2, y=2x.\\$

$x^2-2x=0, x(x-2)=0,\\ x_1=0, x_2=2.\\ V=\pi \int^2_0 (4x^2-x^4)dx=\\ =\pi(\frac{4}{3}x^3-\frac{1}{5}x^5)|^2_0=\\ =\pi(\frac{4}{3}2^3-\frac{1}{5}2^5)=\\ =\pi(\frac{32}{3}-\frac{32}{5})=\\ =32\pi(\frac{1}{3}-\frac{1}{5})=\\ =\frac{64\pi}{15}=13.40$