## 3psilon Group Title rotate the region bounded by y = 6 - 2x - x^2 and y = x+6 about the line y=3 one year ago one year ago

1. 3psilon Group Title

My answer differs from that of my teacher's. Can someone just show me the integral

2. zepdrix Group Title

Can you provide the teacher's answer a sec? I wanna see if I made some terrible mistake before I post this integral :P

3. 3psilon Group Title

ok ok

4. 3psilon Group Title

5. 3psilon Group Title

Problem 21 part D

6. zepdrix Group Title

Hmm this is setup really strangly... Clearly the parabola is our upper function. So why is the parabola being subtracted from the line... This is what I came up with, $\large \pi \int\limits_{-3}^0 \left[(6-2x-x^2)-3\right]^2-\left[(x+6)-3\right]^2\;dx$ I should look over my work again though.. bit of a tricky problem. I may have made a mistake somewhere.

7. 3psilon Group Title

But I thought the outer radius was at a height 3 plus the parabola function so why wouldn't it be 3+(6-2x-x^2)?

8. zepdrix Group Title

|dw:1362721421835:dw|So the outer radius $$\large R$$ is going to be $$\large y_1-3$$. Where $$\large y_1=6-2x-x^2$$.

9. zepdrix Group Title

I hope that drawing makes sense... :\ I took a slice and spun it around y=3. And drew some lines to figure out the radius.

10. 3psilon Group Title

It makes sense but can you explain why we subtract 3?

11. zepdrix Group Title

|dw:1362721856820:dw|That new length is the outer radius.

12. zepdrix Group Title

If we were to draw a disk (not hollowed out) with that radius, it would look like this.|dw:1362722006047:dw| This area of this disk, minus the area of, |dw:1362722103066:dw| Gives us the area of our disk hollowed out,\|dw:1362722226356:dw|[\large A=\pi\left[R^2-r^2\right]\]

13. zepdrix Group Title

$\large A=\pi\left[R^2-r^2\right]$

14. zepdrix Group Title

Bah I wish the drawing tool had more functionality :) lol

15. 3psilon Group Title

ohhhh okay okay ! I always had trouble seeing that part! Thank you ! hhha I really appreciate the drawings they helped a lot! Thanks for helping me get ready for my test!

16. zepdrix Group Title

Makes a bit of sense? :D Yay!