Question

Let t be some fixed real number. Find the volume of the region bounded by y =1/x and the y = 0 on the interval [1, t] rotated about the x-axis. What happens as t → ∞?

Answer 1

v = ∫[1,t] πr^2 dx
where r=y=1/x. So,

v = π∫[1,t] 1/x^2 dx
= π(-1/x) [1,t]
= π(1 - 1/t)
as t→∞, v→π