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The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
x = (y − 5)^2, x = 4; about y = 3
I am trying to use the disc method but I am not sure how I would.
 one year ago
 one year ago
The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. x = (y − 5)^2, x = 4; about y = 3 I am trying to use the disc method but I am not sure how I would.
 one year ago
 one year ago

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Dido525Best ResponseYou've already chosen the best response.0
Anyone want to help me set up the integral?
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Well I found the intersection points for y which are y=3 and y=7.
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
Good. This is a nice problem. Have you heard of Pappus's Theorem?
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
Too bad. It is much simpler with the Centroid. Another lesson on another day, perhaps. The Whole thing: \(\pi\int\limits_{0}^{4}(y_{1}3)^{2}\;dx\) The Extra thing: \(\pi\int\limits_{0}^{4}(y_{2}3)^{2}\;dx\) Do you see the nature of the problem that requires two definitions for 'y'? It's NOT a function!
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Yeah, I realized that.
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
Perfect. What are the two definitions?
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Wait you lost me. Never mind.
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
\(x = (y5)^{2}\) Solve for \(y\). You should get TWO results.
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Wait. WHat am I doing...
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
?? Where did the \(x\) go? \(\sqrt{x} = y5\;for\;y>5\) \(\sqrt{x} = 5y\;for\;y<5\) Keep solving!
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Sorry. I had a stupid moment.
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
Super. You almost beat me to it. Now, which one is on top and which one is on the bottom? These are my little yfunctions.
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Top function is \[\sqrt{x}+5\] Bottom function is \[5\sqrt{x}\]
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
The Whole thing: \(\pi\int\limits_{0}^{5}((\sqrt{x}+5) −3)^{2}\;dx\) The Extra part: Oh, you do that one!
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Woudn't it be from 0 to 4?
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
Yes. I just spotted that typo. Sorry about that. Good eye!
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
And the answer is????????
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
136/3 which is wrong :( .
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
That's the whole thing. You have to subtract the small piece. That's why we had two integrals.
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
So it would be:\[\pi \int\limits_{0}^{4}((\sqrt{x}+5)3)^2((5\sqrt{x})3)^2dx\]
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
That should do it. Or, you can do them as separate integrals.
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
THanks so much! Lets see if I get the right answer...
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Yeah it's right :) . Thanks so much!
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
I know. I did it out on maple.
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
Interestingly, if you do a little algebra, you can do a whole lot less calculus. That wonderful integrand you created simplifies to \(8\sqrt{x}\).
 one year ago

tkhunnyBest ResponseYou've already chosen the best response.1
Well, for future reference, if the ENTIRE area is on one side of the axis of rotation, you can pull Pappus' Theorem out of your hat! Good luck when the time comes!
 one year ago
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