## anonymous 4 years ago Can anyone prove the volume of an ellipse?

1. anonymous

Do you mean find an area or actually prove the formula?

2. anonymous

LOL can u read it?

3. anonymous

yeah i can zoom in. so basically they want you to use calculus to derive the formula for volume of a 3-d ellipse

4. anonymous

yes :D

5. anonymous

ok, start with equation of ellipse: $\frac{x^{2}}{b^{2}} +\frac{y^{2}}{a^{2}} = 1$ |dw:1327556444323:dw| then integrate from -b to b, of A(x) which is the area of each circular cross-section where the radius is y: solve the ellipse equation for y $y = \sqrt{a^{2} - \frac{a^{2}}{b^{2}}x^{2}} = \frac{a}{b}\sqrt{b^{2}-x^{2}}$

6. anonymous

Thanks for ur help :D That was awesome and very clear

7. anonymous

wait how did u get that equation

8. anonymous

which one?

9. anonymous

like from the first to the second

10. anonymous

ohhh i get it now

11. anonymous

but do i have to integrate it?

12. anonymous

or is this the end result?

13. anonymous

sorry i skipped some simplifying steps.. no you still have to integrate $V = \pi \int\limits_{-b}^{b}y^{2} dx$ plug that equation in for y

14. anonymous

huh i doubt i will be able to integrate that on my own

15. anonymous

ok $y^{2} = \frac{a^{2}}{b^{2}}(b^{2}-x^{2})$ factor out the constants $V = \frac{a^{2} \pi}{b^{2}}\int\limits_{-b}^{b} (b^{2} -x^{2}) dx$

16. anonymous

|dw:1327557477105:dw|

17. anonymous

yep :)

18. anonymous

|dw:1327557564379:dw|

19. anonymous

Is that correct?

20. anonymous

not quite, you skipped a step first evaluate from -b to b by plugging them in for x

21. anonymous

we don't want x in our answer

22. anonymous

oh ya whoops lol

23. anonymous

|dw:1327557858057:dw|

24. anonymous

|dw:1327557996710:dw|

25. anonymous

This shld be the answer :D

26. anonymous

looks good :) don;t forget about the pi

27. anonymous

oh yes thanks for ur help and patience

28. anonymous

this is the same way of finding volume of sphere as well just make a =b

29. anonymous

ok thanks :D

30. anonymous

Goodnight