# 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.
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)...

greg_rack said:
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
hicetnunc said:
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

## 1. What is the definition of integration when it comes to finding the volume of a solid of revolution around the y-axis?

The process of finding the volume of a solid of revolution around the y-axis involves using the method of cylindrical shells, where the volume is approximated by summing up the volumes of infinitely thin cylindrical shells with height and radius equal to the function value at each point along the y-axis.

## 2. How do you set up the integral for finding the volume of a solid of revolution around the y-axis?

The integral for finding the volume of a solid of revolution around the y-axis is set up by integrating the function with respect to y, and using the limits of integration as the y-values where the solid starts and ends. The integrand is the circumference of the cylindrical shell, which is 2πy times the thickness of the shell.

## 3. Can the method of cylindrical shells be used for any shape revolved around the y-axis?

Yes, the method of cylindrical shells can be used for any shape revolved around the y-axis, as long as the shape can be expressed as a function of y. This includes shapes with holes, as long as the function can account for the hole.

## 4. What is the difference between using the method of cylindrical shells and the disk method for finding the volume of a solid of revolution around the y-axis?

The main difference between the two methods is the way the solid is divided into infinitesimally thin slices. The disk method uses circular disks, while the cylindrical shell method uses cylindrical shells. The disk method is typically used for solids with a circular cross-section, while the cylindrical shell method can be used for any shape.

## 5. Are there any limitations to using the method of cylindrical shells for finding the volume of a solid of revolution around the y-axis?

One limitation of using the method of cylindrical shells is that it can only be used for solids that are revolved around the y-axis. For solids that are revolved around a different axis, a different method, such as the washer method, would need to be used. Additionally, the function must be continuous and positive for the entire range of y-values in order for the method to work accurately.

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