Evaluation of integral having trigonometric functions

WMDhamnekar
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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:
1654415829519.png


1654415851473.png


But when we plot sin((x + y)/2)*cos((x-y)/2), we get the below graph:
1654416090367.png

J(u,v)=|-2| = 2, So, now how to compute integration limits?
 
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I computed the answer as ## \displaystyle\int_0^{0.5}\displaystyle\int_0^{1.5} 2*sin(u)*cos(v)du dv = 0.891024635191 ## . But it is wrong. Correct answer provided is ## 1- \frac{sin(2)}{2} = 0.545351286587##

How is that? What is wrong with my answer?
 
WMDhamnekar said:
I computed the answer as ## \displaystyle\int_0^{0.5}\displaystyle\int_0^{1.5} 2*sin(u)*cos(v)du dv = 0.891024635191 ## . But it is wrong. Correct answer provided is ## 1- \frac{sin(2)}{2} = 0.545351286587##

How is that? What is wrong with my answer?

How did you arrive at those limits? You started with a triangle in the (x,y) plane. Does it seem sensible that you would end up with a rectangle in the (u,v) plane?
 
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pasmith said:
How did you arrive at those limits? You started with a triangle in the (x,y) plane. Does it seem sensible that you would end up with a rectangle in the (u,v) plane?
1654443340888.png
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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