Question

Volumes of Revolution help.

Find the volume formed by rotating the region enclosed by:
y=x^2, x=y^2, about the line x=-2

Answer 1

Find your points of intersection so you know the limits of your integration.

Answer 2

Would they be 0 and 1?

Answer 3

Perfect.

Decide, Shells or Disks? It's about the same difficulty on this nice region. Just pick one.

Answer 4

Are you struggling with the decision? Really, just flip a coin. Maybe both ways?

Answer 5

Shell than the outter equation would be x^2 +2 and inner sqrt(x) +2?

Answer 6

i got the +2 just becuase of the x=-2 thats its flipping around, i dont know if its right

Answer 7

Shells it is, then...

\(2\pi\int\limits_{0}^{1} Radius\cdot Height\; d(Radius)\)

"outer" and "inner" don't mean the same thing for Shells. You just need to subtract them to get the height of each shell.

Answer 8

The "Radius" inside the integral is simply (-2-x). Do you see this?

This gives: \(2\pi\int\limits_{0}^{1}(-2-x)(Height)\;dx\)

Answer 9

yeah this is making a lot of sense actually

Answer 10

Excellent. All we've left is the height of each shell. What say you?

Answer 11

so would that be the difference of sqrtx and x^2?

Answer 12

That's correct! Get them in the right order.

Answer 13

so its (x^2 -sqrtx) and than the integral of that times the radius?

Answer 14

You have it! Let's see what you get.

Answer 15

2pi(2-(1/2)(1/3)(1-2(1)^(3/2)))

Answer 16

its not right though

Answer 17

Hmmm... What did your original integrand look like? It should be \((-2-x)(x^{2}-\sqrt{x})\). Did we get that far?

Answer 18

Yes

Answer 19

That expands to a mess: \(2x^{1/2} - x^{3} - 2x^{2} + x^{3/2}\)

Did you get all that?

Answer 20

yeah let me try it again real quick i may have made a little mistake

Answer 21

i integrated it to this (2 x^(5/2))/5+(4 x^(3/2))/3-x^3 than plugged in 1 and multiplied it by 2pi. Its still wrong though

Answer 22

What's up with "-x^3"? You should have -(1/4)x^4 - (2/3)x^3

Answer 23

-2pi((2 (1)^(5/2))/5+(4 (1)^(3/2))/3-(1/4)(1)^4 - (2/3)(1)^3)
Is this what its supposed to look like?

Answer 24

without the - in the front

Answer 25

Its right, i was off with my integration, thank you a ton for your help!

Answer 26

That looks like it!

Answer 27

It's a lot to wade through. GREAT WORK!!