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Find the volume of a solid using both the disk/washer and shell method.

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!

Re: Find the volume of a solid using both the disk/washer and shell method.

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 ?

Re: Find the volume of a solid using both the disk/washer and shell method.

Re: Find the volume of a solid using both the disk/washer and shell method.

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.