# Double integral confused

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

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Dick
Homework Helper
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.

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 $$\cos u ={{{\rm e}^{{\rm j} u} + {\rm e}^{-{\rm j} u}}\over{2}}$$, and didn't get -8 either. It's probably just a simple error in the substitution process.

I did this out quickly using $$\cos u ={{{\rm e}^{{\rm j} u} + {\rm e}^{-{\rm j} u}}\over{2}}$$, 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?

$$cos(A + B) = cosAcosB - sinAsinB.$$

Why not just use the sum formula for cosine?

$$cos(A + B) = cosAcosB - sinAsinB.$$
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.

hm, it seems no one answered my question: do i have to change the boundary conditions when doing the u-substitution?

yes, you do. out of curiosity, which u substitution did you use?

Dick
Homework Helper
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.

using the sum formula for cosine i got -2?

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using the sum formula for cosine i got -2?
That looks correct

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.

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.

... 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.

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