## anonymous 5 years ago Double integrals in polar coordinates?? I Know nothing about it :S

1. Owlfred

Hoot! You just asked your first question! Hang tight while I find people to answer it for you. You can thank people who give you good answers by clicking the 'Good Answer' button on the right!

2. anonymous

Sometimes integrals are easier to evaluate using polar coordinates (r, θ), where x = r cos θ and y = r sin θ. that is the basic idea behind it

3. anonymous

aha,, and how I can apply it ? can you give me an example ?

4. anonymous

I am not that good with it, just looking at my lecture notes, but I can try

5. anonymous

The height of the hemisphere about the x-y-plane is given by z =sqrt(1 − x^2-y^2) Find the volume under the unit hemisphere with positive z-coordinate and positive y-coordinate.

6. anonymous

region of integration is 1> x^2+y^2 (as it has to be + under sqrt) and y>=0

7. anonymous

now first challenge is to change the region of integration

8. anonymous

aha...

9. anonymous

x^2+y^2 in polar is =r^2 (as sin^2 +cos^2=1)

10. anonymous

so r has to be between 0 and 1

11. anonymous

y>0 implies that rsinθ>0 r is always >0 here (0 to 1) so sinθ has to be >0 implies θ is from 0 to pi clear so far?

12. anonymous

aha ..

13. anonymous

$\int\limits_{0}^{\pi}\int\limits_{0}^{1}\sqrt{1-r ^{2}} drd \theta$

14. anonymous

so this is what we have to do now

15. anonymous

it is not, because of the change of integral dxdy becomes r drdθ

16. anonymous

$rsqrt{(1-r ^{2})}$ this is what you have to integrate

17. anonymous

do you know how?

18. anonymous

one minute, hoe it comes r from 0 to 1 ,, since r^2<1

19. anonymous

how**

20. anonymous

1-x^2-y^2 has to be + as it is under sqrt x^2+y^2= (rcosθ)^2+(rsinθ)^2=r^2 so 1-r^2 has to be +

21. anonymous

1-r^2>0 1>r^2 -1<r<1 ??

22. anonymous

u are right but the example says: positive z-coordinate and positive y-coordinate.

23. anonymous

ahaaa I get it

24. anonymous

25. anonymous

Thanks alot I understood the idea

26. anonymous

cool!

27. anonymous

I have another ques. ?

28. anonymous

do you or do you not? :)

29. anonymous

go on

30. anonymous

in another topic

31. anonymous

Im just a 1st year maths student, so my knowledge has limits :)

32. anonymous

but I can try

33. anonymous

how to find the projection of a=(1,3) on b=(4,0,2) ??

34. anonymous

no clue

35. anonymous

post it as a different question

36. anonymous

ok, thanks for trying :)