Question

Find the volume of the solid generated by rotating the region bounded by the curves y=0, y=x^2+x, and x=2 about the y-axis

Answer 1

$V = \int_{0}^{2} 2\pi x*(x^2 +x) dx =$

$= 2\pi \int_{0}^{2} x^3 +x^2 dx = 2\pi [\frac{x^4}{4} + \frac{x^3}{3}] \Big\vert_{0}^{2} =$

$= 2\pi [4+ \frac{8}{3}] = 2\pi [\frac{20}{3}] = \frac{40 \pi}{3}$

Answer

$V = \frac{40\pi}{3}$