Transforming Double Integral with Change of Variables

[Furbishkov]

Homework Statement

Evaluate the integral

∫∫sin(x+y)/(x+y) dydx over the region D

whereD⊆R² is bounded by x+y=1, x+y=2, x-axis, and y-axis.

Homework Equations

The Attempt at a Solution

I think that I need to use a change of variables but can not find any change of variables that work. One thing I thought would work is using u = sin(x+y) and v = (x+y) but finding the jacobian doesn't work with the u transformation.
This leads me to think I might need to do a taylor expansion instead.

Any help with a recommended change of variables or a step in the right way would be appreciated.

 

[haruspex]

One thing I thought would work is using u = sin(x+y) and v = (x+y)

Try something much simpler, but along similar lines.

 

[vela]

Did you make a sketch of D? If not, do so. Add to it lines of constant v=x+y to see why this is a good choice. That might give you an idea of what to use for u.

 

[Furbishkov]

I sketched D and put in the constant v=x+y but still can not see any good change of variable for u. The sin is really causing me problems. I constantly think I need to make my "u' have the sin term in it but can not come up with anything that works

 

[haruspex]

I sketched D and put in the constant v=x+y but still can not see any good change of variable for u. The sin is really causing me problems. I constantly think I need to make my "u' have the sin term in it but can not come up with anything that works

Given the substitution v=x+y, what is the most obvious way to define u? (It does not involve a trig function.)

 

[vela]

You don't want the sine in the transformation. That's what haruspex was implying when he said to try something simpler.

 

[Furbishkov]

1. Given the substitution v=x+y, what is the most obvious way to define u? (It does not involve a trig function.)

2. Looking at my graph is seems to be convenient if I define u to be a constant, possibly u = 2? ... When I learned change of variables I was told I would never need to come up with the change of variable myself so that is why I'm having some troubles. I appreciate the help.

 

[haruspex]

1. Looking at my graph is seems to be convenient if I define u to be a constant, possibly u = 2? ... When I learned change of variables I was told I would never need to come up with the change of variable myself so that is why I'm having some troubles. I appreciate the help.

What expression for u would result in the u, v coordinate system being Cartesian?

 

[Furbishkov]

What expression for u would result in the u, v coordinate system being Cartesian?

I still am completely stuck. Any way I look at it I get stuck with something I can't integrate with the sin in the integral. I am working off the assumption that v = x+y is correct. This would make my integral something like ∫ sin(v)/v . My last thought is to make u = 1/x+y. But this still seems to give me some integral that I can't solve...

 

[haruspex]

I still am completely stuck. Any way I look at it I get stuck with something I can't integrate with the sin in the integral. I am working off the assumption that v = x+y is correct. This would make my integral something like ∫ sin(v)/v . My last thought is to make u = 1/x+y. But this still seems to give me some integral that I can't solve...

Isn't u=y-x the most obvious choice, by a mile?
If you think that doesn't help, integrate wrt u first, making sure you get the limits right. They depend on v.

 

[Furbishkov]

That change of variable makes sense when I look on it on a graph, thanks. Now for my bounds I get, 1≤v≤2 and -v≤u≤v . I worked out the integral to just become sin(v). Thanks for the help!