Question

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.

Answer 1

Anyone want to help me set up the integral?

Answer 2

Anyone? :( .

Answer 3

Well I found the intersection points for y which are y=3 and y=7.

Answer 4

Good. This is a nice problem. Have you heard of Pappus's Theorem?

Answer 5

No.

Answer 6

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!

Answer 7

Yeah, I realized that.

Answer 8

Perfect. What are the two definitions?

Answer 9

Wait you lost me. Never mind.

Answer 10

@tkhunny

Answer 11

\(x = (y-5)^{2}\)

Solve for \(y\). You should get TWO results.

Answer 12

yeah y=y-5 and y=5-y

Answer 13

Wait. WHat am I doing...

Answer 14

Braid dead lol.

Answer 15

?? Where did the \(x\) go?

\(\sqrt{x} = y-5\;for\;y>5\)

\(\sqrt{x} = 5-y\;for\;y<5\)

Keep solving!