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- Homework Statement
- Evaluate ## \displaystyle\iint\limits_R \sin{(\frac{x+y}{2})} \cos{(\frac{x-y}{2})}dA,## where R is the triangle with vertices (0,0), (2,0) and (1,1).

- Relevant Equations
- Hint: Use the change of variables ##u=\frac{x+y}{2}, v= \frac{x-y}{2}##

R is the triangle which area is enclosed by the line x=2, y=0 and y=x.

Let us try the substitution ##u = \frac{x+y}{2}, v=\frac{x-y}{2}, \rightarrow x=2u-y , y= x-2v \rightarrow x= 2u-x + 2v \therefore x= u +v##

## y=x-2v \rightarrow y=2u-y-2v, \therefore y=u- v## The sketch of triangle is as follows:

But when we plot sin((x + y)/2)*cos((x-y)/2), we get the below graph:

J(u,v)=|-2| = 2, So, now how to compute integration limits?

Let us try the substitution ##u = \frac{x+y}{2}, v=\frac{x-y}{2}, \rightarrow x=2u-y , y= x-2v \rightarrow x= 2u-x + 2v \therefore x= u +v##

## y=x-2v \rightarrow y=2u-y-2v, \therefore y=u- v## The sketch of triangle is as follows:

But when we plot sin((x + y)/2)*cos((x-y)/2), we get the below graph:

J(u,v)=|-2| = 2, So, now how to compute integration limits?