Finding volume of region: y=1/x, y=0, x=1, x=3, abt y-axis

[Jones13]

What is the volume of the solid obtained when the region bounded by the curves y=1/x, y=0, x=1, and x=3 is rotated about the y-axis?

Here's my work:
a(x)=?(1/x)^2 = 1/(?(x^2))

Volume = integral from 1 to 3... 1/(?(x^2))dx
= ?[-1/x]from 1 to 3.
= ?[-1/3-(-1)]
= ?(-1/3 + 1)
= 2?/3 UNITS^2

Teacher told me it was wrong but I don't understand. Thanks.

 

[Dr. Flim-Flam]

Re: Finding the volume of a region

Shell Method:

2? times integral from (1 to 3) [dx] = 4?

Disc Method:

? times integral from (1/3 to 1) (1/y²-1)dy + ? times integral from (0 to 1/3)(8dy) = 4?.

Shell Method is preferable.

 

[Jones13]

Great!

Thanks. I tried the disc method and that's where I got lost. I forgot to find the answer in terms of y rather than x. Again, thanks.