Question

The region bounded by y=5/x, y=0, x=1, and x=3 is rotated about the x-axis. find the volume of the resulting solid.

Answer 1

Volume of Revolution  = $\pi \int\limits^b_a {y^{2}} \, dx$
Volume of Revolution  = $\pi \int\limits^3_1 { (\frac{5}{x})^{2}} \, dx$
= $\pi \int\limits^3_1 ({\frac{25}{x^{2}}) \, dx$
= $25\pi \int\limits^3_1 ({\frac{1}{x^{2}}) \, dx$
= $25\pi \int\limits^3_1 ({x^{-2}}) \, dx$
= $25\pi \ ^{3} [-x^{-1}] _{1}$
$= 25\pi\ ((-1/3) - (-1/1)) \\ = 25\pi\ (2/3) \\ = 50/3 \pi\ units^{3}$