Medal** Please help Interchange the order of the integration and evaluate the integral

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Medal** Please help Interchange the order of the integration and evaluate the integral

Mathematics
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\[\int\limits_{0}^{pi}\int\limits_{\pi}^{\pi} \frac{ siny }{ y }dydx\]
Are those the right intervals?
yes

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Ok, think of it this way |dw:1436942548336:dw|
oki...how do i change my limits
Well we have to draw it out, but as it's continuous we can say \[\int\limits_{a}^{b} \int\limits_{c}^{d} f(x,y) dy dx = \int\limits_{c}^{d} \int\limits_{a}^{b} f(x,y) dxdy\]
ok so my limits flip?
Sec, I don't think it's continous
|dw:1436943113232:dw| this is how your graph would like right
yes it would
Ok, so now that should be easy to interchange, |dw:1436943501108:dw|
only my second integral changes right?
Well, we should get \[\int\limits_{0}^{\pi} \int\limits_{0}^{y} \frac{ \sin y }{ y } dx dy\]
why isnt it o to x?
i thought it would be from x to pi
Notice how x is horizontal, also it wouldn't make sense when you are integrating to have x after right?
right because its dy dx
I'm a bit skeptical about the graph mhm
Maybe I'm just a bit tired haha, well does it look good to you?
i understand what your doing
i would intregate siny/y from o to y first?
Yup
would i do quotient rule to intregrate?
What do you mean?
Like derivatives?
yeah im a bit confused on how to integrate siny/y
No, remember we are integrating respect to x, so we treat siny/y as a constant
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
ok thank you...
|dw:1436944319331:dw|
\[\int\limits_{\pi}^{\pi}f(x)\,dx = 0\] am i missing somthing here ?
Yeah, exactly, but I wasn't really sure whether the interchange should make a difference or not
it shouldn't because we must get the same answer either way
are the limits 0 to y. correct then?
No the limits are wrong, I knew it...something seemed fishy
how to find them? drawing again?
would it be pi to x?
The problem itself is messed up, please double check the limits in original question
i just checked again...those are th limits given in the practice test
\[\int\limits\limits_{0}^{\pi} \int\limits\limits_{x}^{\pi}\] my drawing is for this
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..
Yeah this problem is a bit weird lol
1 Attachment
because both \(x\) and \(\pi\) look alike typographically
the first question
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.
hmm oki lets go with x...since pi to pi doesnt make sense
yeah i think thats the best we can do
Yeah lets treat it like x, you'll learn this way to I guess
|dw:1436944946903:dw|
alright so our new limits would be in respect to the drawing
So go ahead and integrate what we originally had
\[\large \int\limits\limits_{0}^{\pi} \int\limits\limits_{0}^{y} \frac{ \sin y }{ y } dx dy\]
siny/y changes to siny/y *y= siny?
Mhm?
i got 1 for the first integration
\[\large \int\limits_{0}^{\pi}\int\limits_{x}^{\pi} \frac{ \sin y }{ y }\,dydx = \int\limits_{0}^{\pi}\int\limits_{0}^{y} \frac{ \sin y }{ y }\,dxdy = \int\limits_0^{\pi} \frac{\sin y}{y}*y \,dy = -2\]
not just 2?
|dw:1436945173549:dw|
Make sense?
yes thank you so much!
and that should = 2 as ganeshie showed
Ahh right, my mistake, it should be +2
Yeah haha
wow thank you so much you guys!
Np, you can finish it off right to get 2?
yes :)
Ok cool
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
yes i agree
Alright cool, good luck on your exam :)

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