Find volume of solid generated (Calc 2 possibly involving integration by parts)

[lovex24]

[solved]Find volume of solid generated (Calc 2)

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y=e^x, and the line x = ln 2 about the line x= ln 2.

So I tried graphing it to see visually, and the expression I got for calculating the volume was ∫π(ln2-lny)^2dy, evaluating from 0 to 2 using disk method, and the answer I got was 4π, but apparently that doesn't match the answer in the back of the book. I'd really appreciate if someone can help me out!!

Off topic: First time posting a thread here, may I ask how do you type the mathematical symbols such as the integral sign and whatnot, or do I have to manually copy and paste from other website?

 

[daon2]

Your region must be broken into two pieces if you want to use the "washer" method. Notice the function farthest to your axis changes when y=1.

For 0<=y<=1

\(\displaystyle \displaystyle \pi \int_0^1 (\ln 2)^2 dy\)

For 1< y<=2

\(\displaystyle \displaystyle \pi \int_1^2 (\ln 2 -\ln y)^2 dy\)

And you add them.

If you use the cylindrical shells method, it is

\(\displaystyle \displaystyle 2\pi \int_0^{\ln 2} (\ln(2)-x)e^{x} dx\)

 

[Deleted member 4993]

daon posted the above post using LaTex.

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[lovex24]

Your region must be broken into two pieces if you want to use the "washer" method. Notice the function farthest to your axis changes when y=1.

For 0<=y<=1

\(\displaystyle \displaystyle \pi \int_0^1 (\ln 2)^2 dy\)

For 1< y<=2

\(\displaystyle \displaystyle \pi \int_1^2 (\ln 2 -\ln y)^2 dy\)

And you add them.

If you use the cylindrical shells method, it is

\(\displaystyle \displaystyle 2\pi \int_0^{\ln 2} (\ln(2)-x)e^{x} dx\)

thanks so much for finding the error I had!!! I was able to match the answer to the back of the book!

 

[lovex24]

daon posted the above post using LaTex.

This web-site supports LaTex.

Google "LaTex" to learn more about it.

Thanks a lot for the info! I'll look into it as soon as I finish my hw!