Question: I want to find $$\int \int\_ R \sin\left(\frac{x-y}{x+y}\right)dA$$ where $R$ is the trapezoid with vertices $(1,1)$, $(2,2)$, $(4,0)$ and $(2,0$). I've seen some examples with similar integrals in my calculus book using triangles instead, but I'm confused as to what happens when we decide to use some other geometric figure. The way I understood it from these examples was that you integrate $dx$ from the lowest x values to the highest (where y=0) ($2$ and $4$ here) , and for the other integral the bounds $x=1$ from $(1,1)$ and $-x+y$ (the line through $(1,1)$ and $(2,2)$

Thus I get $$\int\_2^4 dx \int\_1^{-x+y}\sin\left(\frac{x-y}{x+y}\right)dy$$. I'm not sure if my understanding of these bounds are correct though as I didn't succeed in trying to evaluate this with wolfram.

Appreciate any help!

Answer: 
Since you have to have $1$ blue and $1$ yellow marble, the problem comes down to finding the number of different subsets of: $3$ red, $1$ blue, $3$ yellow, and $1$ white marble

Now, you can choose $0,1,2$, or $3$ red marbles, so $4$ possibilities there. Same for the yellow ones of course. Then, you can either add a blue or not, so $2$ options there, and same for the white.

So, that's $4 \cdot 4 \cdot 2 \cdot 2= 64$ possible subsets.
Use generating functions. Subtracting 1 blue and 1 yellow, the generating function here is $(1+x)^2(1+x+x^2+x^3)^2$ or $x^8 + 4 x^7 + 8 x^6 + 12 x^5 + 14 x^4 + 12 x^3 + 8 x^2 + 4 x + 1$. Adding the coefficients gives the answer, i.e. 64.