## anonymous one year ago Medal** Please help Interchange the order of the integration and evaluate the integral

1. anonymous

$\int\limits_{0}^{pi}\int\limits_{\pi}^{\pi} \frac{ siny }{ y }dydx$

2. Astrophysics

Are those the right intervals?

3. anonymous

yes

4. Astrophysics

Ok, think of it this way |dw:1436942548336:dw|

5. anonymous

oki...how do i change my limits

6. Astrophysics

Well we have to draw it out, but as it's continuous we can say $\int\limits_{a}^{b} \int\limits_{c}^{d} f(x,y) dy dx = \int\limits_{c}^{d} \int\limits_{a}^{b} f(x,y) dxdy$

7. anonymous

ok so my limits flip?

8. Astrophysics

Sec, I don't think it's continous

9. Astrophysics

|dw:1436943113232:dw| this is how your graph would like right

10. anonymous

yes it would

11. Astrophysics

Ok, so now that should be easy to interchange, |dw:1436943501108:dw|

12. anonymous

only my second integral changes right?

13. Astrophysics

Well, we should get $\int\limits_{0}^{\pi} \int\limits_{0}^{y} \frac{ \sin y }{ y } dx dy$

14. anonymous

why isnt it o to x?

15. anonymous

i thought it would be from x to pi

16. Astrophysics

Notice how x is horizontal, also it wouldn't make sense when you are integrating to have x after right?

17. anonymous

right because its dy dx

18. Astrophysics

I'm a bit skeptical about the graph mhm

19. Astrophysics

Maybe I'm just a bit tired haha, well does it look good to you?

20. anonymous

21. anonymous

i would intregate siny/y from o to y first?

22. Astrophysics

Yup

23. anonymous

would i do quotient rule to intregrate?

24. Astrophysics

What do you mean?

25. Astrophysics

Like derivatives?

26. anonymous

yeah im a bit confused on how to integrate siny/y

27. Astrophysics

No, remember we are integrating respect to x, so we treat siny/y as a constant

28. Astrophysics

I'm still skeptical about the graph, it's throwing me off, and I'm a bit tired to confirm I'm going to tag the illustrious @ganeshie8 to confirm

29. anonymous

ok thank you...

30. Astrophysics

|dw:1436944319331:dw|

31. ganeshie8

$\int\limits_{\pi}^{\pi}f(x)\,dx = 0$ am i missing somthing here ?

32. Astrophysics

Yeah, exactly, but I wasn't really sure whether the interchange should make a difference or not

33. ganeshie8

it shouldn't because we must get the same answer either way

34. anonymous

are the limits 0 to y. correct then?

35. Astrophysics

No the limits are wrong, I knew it...something seemed fishy

36. anonymous

how to find them? drawing again?

37. anonymous

would it be pi to x?

38. ganeshie8

The problem itself is messed up, please double check the limits in original question

39. anonymous

i just checked again...those are th limits given in the practice test

40. Astrophysics

$\int\limits\limits_{0}^{\pi} \int\limits\limits_{x}^{\pi}$ my drawing is for this

41. ganeshie8

as you can see the original integral, as-it-is, evaluates to 0. end of story. take a screenshot and attach if you want to further debug for a possible typo..

42. Astrophysics

Yeah this problem is a bit weird lol

43. anonymous

44. ganeshie8

because both $$x$$ and $$\pi$$ look alike typographically

45. anonymous

the first question

46. Astrophysics

Mine works if you have an x, I'm going to stick with it being x haha, the other does not make sense, now I think that is an x instead of pi.

47. anonymous

hmm oki lets go with x...since pi to pi doesnt make sense

48. ganeshie8

yeah i think thats the best we can do

49. Astrophysics

Yeah lets treat it like x, you'll learn this way to I guess

50. Astrophysics

|dw:1436944946903:dw|

51. anonymous

alright so our new limits would be in respect to the drawing

52. Astrophysics

53. Astrophysics

$\large \int\limits\limits_{0}^{\pi} \int\limits\limits_{0}^{y} \frac{ \sin y }{ y } dx dy$

54. anonymous

siny/y changes to siny/y *y= siny?

55. Astrophysics

Mhm?

56. anonymous

i got 1 for the first integration

57. ganeshie8

$\large \int\limits_{0}^{\pi}\int\limits_{x}^{\pi} \frac{ \sin y }{ y }\,dydx = \int\limits_{0}^{\pi}\int\limits_{0}^{y} \frac{ \sin y }{ y }\,dxdy = \int\limits_0^{\pi} \frac{\sin y}{y}*y \,dy = -2$

58. anonymous

not just 2?

59. Astrophysics

|dw:1436945173549:dw|

60. Astrophysics

Make sense?

61. anonymous

yes thank you so much!

62. Astrophysics

and that should = 2 as ganeshie showed

63. ganeshie8

Ahh right, my mistake, it should be +2

64. Astrophysics

Yeah haha

65. anonymous

wow thank you so much you guys!

66. Astrophysics

Np, you can finish it off right to get 2?

67. anonymous

yes :)

68. Astrophysics

Ok cool

69. Astrophysics

Yeah about earlier, the question wasn't really making sense to me, because when we have such integrals $\int\limits_{x}^{x} dx = 0$ it makes sense right using FTC

70. anonymous

yes i agree

71. Astrophysics

Alright cool, good luck on your exam :)