anonymous 4 years ago Is this setup correct? Q: "Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis." x = 1 + y$$^2$$, x = 0, y = 1, y = 3 $\huge\int\limits_0^{\sqrt{2}} (2 \pi)(y)((3) - (1+y^2)) dy$ y=3 is the right function x=1+y$$^2$$ is the left function right - left Graph below:

1. anonymous

|dw:1341360986379:dw| Look good @KingGeorge ? :-)

2. KingGeorge

I think you switched some things around. First of all, the lines you drew for y=1,3 are vertical lines when they should be horizontal. In short, check the area you want integrate again.

3. anonymous

Wait a minute I'm getting zero when I evaluate...

4. anonymous

Oh... I think believe I see what I did there, let's try that graph again: |dw:1341361482202:dw|

5. KingGeorge

That looks better. So your new integrand and bounds would be...?

6. anonymous

$\huge\int\limits_1^3 (2 \pi)(y)((3) - (1+y^2)) dy = -\frac{26}{3}$ The shells are growing out from the x-axis but I'm still missing something, a radius change perhaps? But which one? +1 or -1?

7. anonymous

Seems like 1-y is logical for a shift upward, the others evaluate negative

8. KingGeorge

The only change I would make is in the integrand again. The right side is now $$(1+y)^2$$ and the left is 0. Hence, you should have $\huge\int\limits_1^3 (2 \pi)(y)( (1+y^2)-0) dy$

9. anonymous

$\large \int\limits_1^3 (2 \pi)(y)( (1+y^2)-0) dy =^? \int\limits_1^3 (2 \pi)(1-y)((3) - (1+y^2)) dy = \frac{44}{3}$

10. anonymous

No wait the first is 44/3 and the second is 44$$\pi$$/3

11. KingGeorge

I think I forgot a close parentheses. $\huge\int\limits_1^3 (2 \pi)(y)( (1+y)^2) dy$

12. anonymous

Just noticed it ran off the page too, sry about breaking margins >_<

13. anonymous

$\huge\int\limits_1^3 (2π)(y)((1+y)^2)dy = \frac{248\pi }{3}$ Seems logical, let me give it a try...

14. anonymous

Alas no, it is incorrect

15. anonymous

Sheesh these particular problems are being, well, problems tonight :D

16. anonymous

*looks back over steps*

17. KingGeorge

Wait, I misread the question. There shouldn't be a close parentheses there XD Have you already tried it without the close parentheses that I added?

18. anonymous

(2π)(y)((1+y)^2)dy <-- literally what I started working with

19. anonymous

Oh it's on the wrong side!

20. KingGeorge

How about $\large\int\limits_1^3 (2 \pi)(y)( 1+y^2) dy?$

21. anonymous

That's a quadratic now by mistake

22. anonymous

And 48π is perfectly correct, awesome!!!

23. anonymous

Comes out very nice in the end, no fractions

24. KingGeorge

Awesome. Now I just need to read the problems correctly.

25. anonymous

Me too lol

26. anonymous

What am I thinking, making vertical lines when it asked for horizontal lol