Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) $. Evaluate the integral :
$$ \iint_D e^{\frac{y-x}{y+x}} $$
a) by transforming to polar coordinates
b) by using the transformation $u = y - x$ and $v = y + x$
In a) i planned to describe the domain $0 \leq r \leq 1$ , $0 \leq \theta \leq \frac{\pi}{2}$ , and then i would use some trigonometric identities to express $\frac{y-x}{y+x}$ , i thought something like : $$y=r\cdot\sin(\theta) \\ x=r\cdot\cos(\theta) $$ would apply. Then my plan would be to go something like this : $$ \int_0^\frac{\pi}{2}e^{\left(\cfrac{\sin(\theta)-\cos(\theta)}{\sin(\theta)+\cos(\theta)}\right)}d\theta\int_0^1e^{\left(\cfrac{-r}{r}\right)}\cdot r\,dr $$
I can evaluate the outer intergral to $\frac{1}{2}\cdot e^{-1}$ But in terms of the inner intergral I do not now which trigonometric rewrite to use is it even correct this far ?? The answer is supposed to be : $$\frac{1}{4}\cdot(e-e^{-1})$$
In terms of b) my plan was to use the change of variables formula : $$ \iint_Df(x,y)dx dy= \iint_Sg(u,v)\Bigg\vert{\frac{d(x,y)}{d(u,v)}\Bigg\vert} du dv $$ I would then use the transformation $u = y - x$ and $v = y + x$, then find the partial differentials : $$ e^{\frac{y-x}{y+x}} dx$$ $$ e^{\frac{y-x}{y+x}}dy$$ $$ e^{\frac{u}{v}}du$$ $$ e^{\frac{u}{v}}dv$$
And then i would set up a matrix like this : $${\frac{d(u,v)}{d(x,y)}}=\cfrac{1}{{\cfrac{d(x,y)}{d(u,v)}}}$$
But the differentions result in some huge expressions and the matrix i am able to set leads to a result looking like this : $$ \frac{ \frac{1}{2}\cdot(y+x)^{4} } {\bigg(e^{\frac{2y-2x}{y+x}}\cdot x\cdot y+x^2+u\cdot y \bigg)} $$ My original plan was to then take the double integral of the result of the matrice in terms of du & dv but i am not able to figure out how to do this with so many unknowns. And upon applying the transformation i am not shure about how the correct parametrization will look like my best gues is :
Original triangle $( 0,0 ) , ( 1,0 )$ and $( 0,1 )$ Upon transformation $( 0,0 ) , ( -1,1 ))$ and $( 0,1 )$
Hope that someone will try to give me a hint i would really like to be able to find the rigth solution : $$ \frac{1}{4}\cdot(e-e^{-1}) $$