I don't see why it would give difficult integrals. Rotating around the x-axis gives disks with radii y and so area . The square root, after squaring, will be gone so you just have integrals of logarithms. What is the integral of ?
Let R be the region bounded by the following curves. Let S be the solid generated when R is revolved about the given axis. If possible, find the volume of S by both the disk/washer and shell methods. Check that your results agree.
I have attached an image with the functions and the correct answers.
I have tried integrating this both as f(x) and f(y), but both lead me to have to take anti-derivatives that seem impossible with the calculus i have learned so far (u-substitution would not work).
I tried to type some of my work here, but when it posted it came out in a bunch of random letters and numbers and such.
Thank you for the help!
I don't see why it would give difficult integrals. Rotating around the x-axis gives disks with radii y and so area . The square root, after squaring, will be gone so you just have integrals of logarithms. What is the integral of ?
For integrals involving logarithms, it's often useful to integrate by parts. For example, to integrate let
and let which gives
Now, to the problem at hand.
Washer Method:
First we determine where intersects the other two curves:
(ignoring the negative root)
So,
Shell Method:
This integral is relatively straightforward. I leave it to you to show that it produces the same result.
Thank you very much to the both of you.
So we haven't learned how to do integration by parts with logarithms and such, so i guess that's where I got stuck, but other than that it's nice to know that my thought process was all correct. I did some research on how you got that and it makes some sense now. (yay)
Much appreciation to the both of you. I'm now a bit more confident with knowledge of this material.