## Moyo30 Group Title Find the volume of the torus (doughnut) formed by rotating the circle x2 + (y − 2)2 = 1 about the x-axis. one year ago one year ago

1. Moyo30 Group Title

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

2. klimenkov Group Title

|dw:1357858021843:dw|

3. Moyo30 Group Title

???

4. klimenkov Group Title

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

5. Moyo30 Group Title

yes.

6. klimenkov Group Title

|dw:1357858353355:dw|

7. Moyo30 Group Title

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

8. klimenkov Group Title

You forgot 2.

9. Moyo30 Group Title

10. klimenkov Group Title

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

11. Moyo30 Group Title

y subtracting?

12. klimenkov Group Title

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

13. klimenkov Group Title

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

14. klimenkov Group Title

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

15. Moyo30 Group Title

|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. Moyo30 Group Title

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 Group Title

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 Group Title

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

19. Moyo30 Group Title

thx

20. klimenkov Group Title

Have you understood something?

21. Moyo30 Group Title

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 Group Title

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. Moyo30 Group Title

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

24. klimenkov Group Title

Why square?

25. Moyo30 Group Title

juss realised my prob no squaring

26. Moyo30 Group Title

integrate, limits the intersection of the lines

27. klimenkov Group Title

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

28. Moyo30 Group Title

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

29. Moyo30 Group Title

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

30. klimenkov Group Title

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

31. Moyo30 Group Title