Question

Find the volume v of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x + 1, y = 0, x = 0, x = 7; about the x-axis

Answer 1

A cross-section is a disc with radius $x + 1$

$V = \int_0^7 A(x)\, dx = \int_0^7 \pi(x+1)^2dx = \int_0^7 \pi(x^2 + 2x + 1)dx \\ \\ = \pi \left[ \frac{1}{3}x^3 + x^2 + x\right]_0^7 = \pi\left( \frac{1}{3}(7)^3 + 7^2 + 7\right) = \frac{511}{3}\pi$