Find the volume of the solid generated by revolving the region about the y axis

Question

The area of the region is 1.

y < or equal to 1/x^2, y> or equal to 0, x > or equal to 1.

Volume of the solid generated by revolving the region about the x axis=pi/3.

Please do not use the Shell method.

Answer 1

All you have to do is integrate π f(x)² from 1 to infinity. As Joel points out, because it is rotated about the x-axis, you use the disk rather than shell method. It's always nice when a solution is provided

Using I[a,b] for the integral from a to b and E[a,b] for evaluation from a to b,

as f(x) = 1/x² , π f(x)² = π (1/x²)² = π/x⁴ ; as x >= 1, we integrate from 1 to ∞.

Then, I[1, ∞) π/x⁴ dx = - π/3x³ E[1, ∞) = 0 - - π/3 = π/3