# Volume of a solid of revolution around the y-axis (def. integration)

• greg_rack
In summary, Mark44's method is much more efficient and easier to understand than my original solution.f

#### greg_rack

Gold Member
Homework Statement
Calculate the volume of the solid generated by rotating around the y-axis the plane region delimited by curves:
##y=e^x##
##x=0##
##x=1##
##y=0##
Relevant Equations
Definite integrals definition
First, I calculated the inverse of ##y=e^x## since we're talking about y-axis rotations, which is of course ##x=lny##.
Then, helping myself out with a drawing, I concluded that the total volume of the solid must've been:
$$V=\pi\int_{0}^{1}1^2 \ dy \ +(\pi\int_{1}^{e}1^2 \ dy \ - \pi \int_{1}^{e}ln^2y \ dy)$$
However this leads me to a wrong result; I must be getting something wrong, probably in the second integration(from 1 to e)...

Homework Statement:: Calculate the volume of the solid generated by rotating around the y-axis the plane region delimited by curves:
##y=e^x##
##x=0##
##x=1##
##y=0##
Relevant Equations:: Definite integrals definition

First, I calculated the inverse of ##y=e^x## since we're talking about y-axis rotations, which is of course ##x=lny##.
Then, helping myself out with a drawing, I concluded that the total volume of the solid must've been:
$$V=\pi\int_{0}^{1}1^2 \ dy \ +(\pi\int_{1}^{e}1^2 \ dy \ - \pi \int_{1}^{e}ln^2y \ dy)$$
However this leads me to a wrong result; I must be getting something wrong, probably in the second integration(from 1 to e)...
If you integrate using shells, the integral is much simpler. With this method, the typical volume element is ##\Delta V = 2\pi \text{radius} \cdot \text{height} \cdot \text{width} = 2\pi x e^x \Delta x##. The shells run from x = 0 to x = 1.

• greg_rack
To me, your equation for the volume ##V## looks fine, so if you don't get the correct solution I think you may have made a mistake later during your evaluation of the integrals. In particular, the integral ##\int_{1}^{e}ln^2y \ dy## looks a little nasty. Have you tried partial integration?

Otherwise, Mark44 gives a nice alternative way to solve for ##V##. That integral might look intimidating at first, but it has the advantage that its solution can usually be found in "cheat sheets".

• greg_rack
To me, your equation for the volume ##V## looks fine, so if you don't get the correct solution I think you may have made a mistake later during your evaluation of the integrals. In particular, the integral ##\int_{1}^{e}ln^2y \ dy## looks a little nasty. Have you tried partial integration?

Otherwise, Mark44 gives a nice alternative way to solve for ##V##. That integral might look intimidating at first, but it has the advantage that its solution can usually be found in "cheat sheets".
Yup, I have managed to solve the integral by parts... and my procedure was actually correct; I only forgot to multiply a term by ##\pi## I gave this problem a chance by mere intuition, without having yet studied definite integrals for such geometrical applications, and I believe that's why the method I came up with looks anything but convenient!
Definitely going to go with @Mark44's :)

Thanks guys

• hicetnunc