## anonymous 3 years ago Find the volume of the torus (doughnut) formed by rotating the circle x2 + (y − 2)2 = 1 about the x-axis.

1. anonymous

Hint: $\int\limits_{-1}^{1}\sqrt{1-x ^{2}}=\frac{ \pi }{ 2 }$

2. klimenkov

|dw:1357858021843:dw|

3. anonymous

???

4. klimenkov

Do you know the formula for the volume of solid rotating $$y=f(x)$$ about $$x$$-axis?

5. anonymous

yes.

6. klimenkov

|dw:1357858353355:dw|

7. anonymous

$\pi \int\limits_{a}^{b}f(x)dx$

8. klimenkov

You forgot 2.

9. anonymous

10. klimenkov

|dw:1357858535828:dw| Subtract the right volume from the left. Got it?

11. anonymous

y subtracting?

12. klimenkov

|dw:1357858823945:dw| It is like subtracting this areas.

13. klimenkov

$$V=\pi\int_{-1}^1(\sqrt{1-x^2}+2)^2dx-\pi\int_{-1}^1(-\sqrt{1-x^2}+2)^2dx$$

14. klimenkov

If you don't get it now just sit and think about it very thoroughly.

15. anonymous

|dw:1357858979556:dw| the graph of x^2+(y-2)^2=1 is something like that how do we get to the penultimate drawing

16. anonymous

i kinda get it you reflect a portion of the circle in the x axis and find the region bounded by the two arcs|dw:1357859345394:dw|

17. klimenkov

if you try to write y depending on x, you will get 2 functions: $$y=\sqrt{1-x^2}+2$$ and $$y=-\sqrt{1-x^2}+2$$. Now think what will be if you rotate them about x-axes. And then use your imagination to unterstand how to get the needed volume. Really the plot is:|dw:1357859397346:dw|

18. klimenkov

But if you are lazy and dishonest student you can just use the formulas I've written above.

19. anonymous

thx

20. klimenkov

Have you understood something?

21. anonymous

yes it is basically a volume but it complicated by the fact that the function that we get after making y the subject of the formula is composed of two equations so we add the two. right?

22. klimenkov

It is very hard to tell if you got something or not. I don't know your level of knowledge. Tell me honestly: can you find the area between $$y=x^2$$ and $$y=1-x^2$$?

23. anonymous

subtract x^2 from 1-x^2 and integrate

24. klimenkov

Why square?

25. anonymous

juss realised my prob no squaring

26. anonymous

integrate, limits the intersection of the lines

27. klimenkov

Yes. The similar situation is here. But now we have a volume to subtract. Hope, you got something, if not - very bad.

28. anonymous

so basically in this prob they ask us to find the volume rather than the area

29. anonymous

can you pls draw the resultant volume if it is not too diff

30. klimenkov

Yes. The volume of the doughnut.|dw:1357861006645:dw|

31. anonymous