## joannaj93 Group Title Using a polar double integral, find the volume in the first octant below the surface z=xy, inside cylinder r=4cos(theta), and between the cylinders x^2 + y^2=4 and x^2 + y^2=8. one year ago one year ago

1. TuringTest Group Title

|dw:1352751755636:dw|

2. TuringTest Group Title

in polar the three graphs are$r=2$$r=2\sqrt2$$r=4\cos\theta$can you find the intersection points in terms of theta?|dw:1352752221119:dw|

3. Algebraic! Group Title

1st octant...

4. joannaj93 Group Title

Make the equations equal to one another...

5. TuringTest Group Title

yes, and what do you get for theta?

6. joannaj93 Group Title

arccos 2^(1/2)/2

7. TuringTest Group Title

well that's one, what angle does that correspond to? and what is the other intersection?

8. joannaj93 Group Title

45 degrees

9. joannaj93 Group Title

and 60 degrees

10. TuringTest Group Title

yes, 45 and 60, but you better use radians in calc|dw:1352754013710:dw|

11. TuringTest Group Title

|dw:1352754159926:dw|

12. TuringTest Group Title

so which fuunction is the outer radius and which one the inner radius for$0\le\theta\le\frac\pi4$?

13. joannaj93 Group Title

r=2 is inner and r=2sqrt2 is outer...

14. TuringTest Group Title

yes, good :) and for the interval$\frac\pi4\le\theta\le\frac\pi3$what is it?

15. joannaj93 Group Title

r=4cos (theta) is outer and r=2sqrt2 is inner...

16. TuringTest Group Title

I think you have the inner wrong|dw:1352754952806:dw|

17. joannaj93 Group Title

oh, inner is r=2

18. TuringTest Group Title

right and what are your integrands?

19. joannaj93 Group Title

I'm not too sure about that

20. TuringTest Group Title

you are integrating xy sub in x=rcost, y=rsint

21. TuringTest Group Title

above t=theta in case you didn't guess also you need to use dA for polar coordinate

22. joannaj93 Group Title

so it'll be the double integral of r^2 cost sint...

23. TuringTest Group Title

that is the xy part, and what is dA in polar?

24. joannaj93 Group Title

dr dt

25. joannaj93 Group Title

sorry, r dr dt

26. TuringTest Group Title

yes, forget that extra r and you will be sorry :P so can you write your integrals yet?

27. joannaj93 Group Title

I dont think so

28. TuringTest Group Title

do you see that it will be two integrals?

29. joannaj93 Group Title

im getting three rs and thethas

30. TuringTest Group Title

there are two regions to consider|dw:1352756391639:dw|what are the bounds for r and theta on the first integral? what are they on the second? those will be the bounds of your integral.

31. joannaj93 Group Title

r=2 to 2√2 and t= 0 to pi/4 and r=2√2 to 4cosθ and t= 0 to pi/3

32. TuringTest Group Title

on the second region the lower bound on r is not 2√2 as we have already discussed, and the lower bound on theta is not 0, but the upper bound of the previous integral

33. joannaj93 Group Title

Sorry about that. give me a sec

34. joannaj93 Group Title

for the second integral: r=2 to 4cost and t= pi/4 to pi/3

35. TuringTest Group Title

good :) so how 'bout setting up those integrals? (two double intergals)

36. joannaj93 Group Title

$\int\limits_{0}^{\pi/4}\int\limits\limits_{2}^{2\sqrt{2}}r ^{2}\cos \Theta \sin \Theta r dr d \Theta$

37. TuringTest Group Title

that's the first one, yes

38. joannaj93 Group Title

$\int\limits_{\pi/4}^{\pi/3} \int\limits_{2}^{4\cos \Theta} r2\cosΘ\sinΘrdrdΘ$

39. TuringTest Group Title

yes, good :) then add them together to find the total voume happy integrating

40. joannaj93 Group Title

Oh, God this does not look like fun, but Thanks for the help :)

41. TuringTest Group Title

It's really not that hard if you can just remember to integrate with respect to theta and r separately, but I'm still too lazy to do it. Welcome!