What is the volume of rotation of; 1/x; from [1,inf) when spun around the x axis ?

Question

amistre64 · Answer

$$\pi \int\limits_{1}^{\infty} [1/x]^2 dx$$

anonymous · Answer

Since the curve is being rotated about the x axis, we can see that the radius of the
inverted funnel so formed will be in the direction of the y axis. 
Now, since the funnel is never ending, it will not terminate at any one point and so we can
take the integral of the area of the base of the funnel from 0 to infinity. This will give us the required volume.
$$\int\limits_{1}^{\infty}  \pi*y ^{2} dx$$$$= \pi \int\limits_{0}^{\infty} (1/x ^{2}) dx$$$$= \pi$$ , when we solve the integral.

anonymous · Answer

1 to infinity in the 2nd integral*

amistre64 · Answer

it is pi :)  but that looks copied and pasted :)  can you expand it out?

anonymous · Answer

what looks copied and pasted? we solve questions similar to this in my calculus class..
integral (1/x^2)  = [-1/x] 

now in the limits 1 to infinity we get [0 - (-1)] * pi

= pi.

amistre64 · Answer

the breaks between the sentences; and the exact wording from the website I saw this on is a dead give away :)

anonymous · Answer

:O

amistre64 · Answer

the solution is the same regardless of where the info is retrieved tho so kudos ;)

anonymous · Answer

lol alright

amistre64 · Answer

$$F(x) = \pi \int\limits_{0}^{\infty} x^{-2} dx \rightarrow F(x)=\pi. \frac{-1}{x}$$

amistre64 · Answer

$$F(\infty) = \pi.0 = 0$$$$F(1) = \pi.-1 = -\pi$$

amistre64 · Answer

$$F(\infty) - F(1) = 0 -(-\pi) = \pi$$