## rainbow22 2 years ago find the volume of the solid formed by revolving the region bounded by y=x^2, y=0, and x=2 about the y axis

1. Thomas9

Do you know about multivariable calculus perhaps? Or do you want the easy-not-understanding-what-you're-doing way

2. rainbow22

3. rainbow22

No, I am supposed to use the Disk Washer method. I can do it with antiderive(2-y) to get the right answer but I'm pretty sure that's not following th formula I'm given which is outer radius minus inner radius.

4. Thomas9

Disk Washer?

5. rainbow22

There's a hole in the solid.. so it's called Washer Method.

6. Thomas9

So how does this method work?

7. rainbow22

I do antiderivative from a-b in terms of x or y (y in this case because I rotate about y axis) pi (outer radius)^2 - (inner radius)^2

8. rainbow22

$\int\limits_{a}^{b} \pi (outer radius)^2 -(innerradius)^2 dx$

9. Thomas9

I don't how that works, I'm afraid.

10. zepdrix

|dw:1359838432884:dw|

11. zepdrix

|dw:1359838561883:dw|So let's look at one slice.

12. zepdrix

Oh you're doing the washer/disk method, my mistake.. Lemme slice that differently.

13. zepdrix

|dw:1359838731527:dw|Ok this type of slice will give us a Disk.

14. zepdrix

|dw:1359838764034:dw|So we want to get the Volume of this disk.

15. zepdrix

The outer radius appears to be the line x=2. And the inner radius our function which is in terms of x, we'll need it in terms of y to integrate since we sliced in the y direction (dy thickness)

16. zepdrix

$\large y=x^2 \qquad \rightarrow \qquad x=\sqrt y$

17. zepdrix

$\large V=\pi\left[(2)^2-(\sqrt y)^2\right]dy$

18. zepdrix

Then to find the total volume in this enclosed area, we Integrate (add up all the slices) from one intersecting point to another. Where do they intersect? Ummm looks like.. y=0 and y=4?

19. zepdrix

Simplifying things down gives us, $\large \pi \int\limits_0^4 4-y \;dy$

20. zepdrix

Imma check my work real quick to make sure I didn't make a mistake somewhere. A little tired, it's possible. lol

21. zepdrix

Yah I think that's right. Confused about any of that? I went through it a little sloppy D:

22. rainbow22

It's right. I realize what I did wrong! I had the wrong intersections/points. I did instead from -2 to 2.. not realizing that it was based on terms of y . thank you so much!

23. zepdrix

Oh cool c: