Double Integral Confusion: How Do I Handle Boundaries with U-Substitution?

[magnifik]

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

 

[Dick]

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.

 

[stevenb]

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.

 

[Raskolnikov]

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?

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

 

[stevenb]

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.

 

[magnifik]

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

 

[Raskolnikov]

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

 

[Dick]

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.

 

[magnifik]

using the sum formula for cosine i got -2?

 

[stevenb]

using the sum formula for cosine i got -2?

That looks correct

 

[magnifik]

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.

 

[stevenb]

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.

 

[stevenb]

... 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.

 

[magnifik]

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