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Double integral confused

  • Thread starter magnifik
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  • #1
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i am confused about the double integral ʃʃ cos(x+2y)dA, where R = [0,pi]x[0,pi/2]
i realize for the integral that i must do u-substitution. when i do this, however, do i also have to change the boundary conditions as in a single integral?

i got -8 without changing the boundary conditions, but i'm not sure if that's right.
i will show my work if anyone wants to check how i got there
 

Answers and Replies

  • #2
Dick
Science Advisor
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The u-substitutions you have to do are pretty easy. But I don't get -8. Maybe you'd better show how you did it.
 
  • #3
699
6
i am confused about the double integral ʃʃ cos(x+2y)dA, where R = [0,pi]x[0,pi/2]
i realize for the integral that i must do u-substitution. when i do this, however, do i also have to change the boundary conditions as in a single integral?

i got -8 without changing the boundary conditions, but i'm not sure if that's right.
i will show my work if anyone wants to check how i got there
I did this out quickly using [tex] \cos u ={{{\rm e}^{{\rm j} u} + {\rm e}^{-{\rm j} u}}\over{2}}[/tex], and didn't get -8 either. It's probably just a simple error in the substitution process.
 
  • #4
193
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I did this out quickly using [tex] \cos u ={{{\rm e}^{{\rm j} u} + {\rm e}^{-{\rm j} u}}\over{2}}[/tex], and didn't get -8 either. It's probably just a simple error in the substitution process.
Why not just use the sum formula for cosine?

[tex] cos(A + B) = cosAcosB - sinAsinB. [/tex]
 
  • #5
699
6
Why not just use the sum formula for cosine?

[tex] cos(A + B) = cosAcosB - sinAsinB. [/tex]
Just used the first thought that came to my mind. It is so simple by either method, and I didn't bother to consider all methods to find the best. I just wanted to get a number to help the OP know if he was right or wrong. Actually, Dick beat me too it, so I just provided a second verification.
 
  • #6
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hm, it seems no one answered my question: do i have to change the boundary conditions when doing the u-substitution?
 
  • #7
193
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yes, you do. out of curiosity, which u substitution did you use?
 
  • #8
Dick
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hm, it seems no one answered my question: do i have to change the boundary conditions when doing the u-substitution?
Integrate dx first and dy second, if that's your choice of order. You'll need a u-substitution for each, and yes, you may need a change of limits. Depending on how you do it. It's kind of useless to debate this until you show how you arrived at the wrong answer.
 
  • #9
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using the sum formula for cosine i got -2?
 
Last edited:
  • #10
699
6
using the sum formula for cosine i got -2?
That looks correct
 
  • #11
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on a somewhat unrelated note, is ln2 the correct answer for ʃʃ (xe^x)/y dydx for R = [0,1]X[1,2]? i did integration by parts for the xe^x part but had the same issue of whether or not to change the start/end point.
 
  • #12
699
6
on a somewhat unrelated note, is ln2 the correct answer for ʃʃ (xe^x)/y dydx for R = [0,1]X[1,2]? i did integration by parts for the xe^x part but had the same issue of whether or not to change the start/end point.
Looks correct to me.
 
  • #13
699
6
... but had the same issue of whether or not to change the start/end point.
Yes, this seems to be a question in your mind. The simple rule is that you only need to change limits if you have a substitution that results in a change of variables. If your functions are still using x and y, and your integration is still over dx and dy, then there is no need to think about changing limits. However, if you change variables, such as u=2x, and/or w=sin(y), then you will generally need to change the limits.
 
  • #14
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Yes, this seems to be a question in your mind. The simple rule is that you only need to change limits if you have a substitution that results in a change of variables. If your functions are still using x and y, and your integration is still over dx and dy, then there is no need to think about changing limits. However, if you change variables, such as u=2x, and/or w=sin(y), then you will generally need to change the limits.
thank you
 

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