How to calculate a double integral

I’ve got the following integral $$\int\int _D \frac{dxdy}{x+y}$$

D is the region bounded by $$x+y = 1, x+y = 4, y=0, x=0$$ and I have to use the transformation $$x = u-uv, y=uv$$

Anyone know what domain to use and how to calculate the integral?

• What did you try? – StubbornAtom May 22 at 14:14
• Well the first step is to find the domain but I couldnt figur that one out, If I get the domain I think I will be able to solve it. @StubbornAtom – Hello there May 22 at 14:16
• – StubbornAtom May 25 at 18:55

$$x = u -uv$$ , $$y=uv$$

$$x_u = 1-v$$ , $$x_v = -u$$

$$y_u = v$$ , $$y_v = u$$

$$J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} 1-v & -u \\ v & u \end{vmatrix} = u -uv + uv = u$$

So, $$dxdy = |J|dudv = u\ du\ dv$$

For regions,

$$x+y = 1 \implies u =1$$

$$x+y = 1 \implies u =4$$

$$y = 0 \implies uv =0 \implies u =0$$ or $$v=0$$

$$x = 0 \implies u - uv=0 \implies u= 0$$ or $$v = 1$$

Can you take it from here?

• Alright I understand the jacobian part, but could you specify the domain in a more generalized way like $D(x,y) | a<x<b, c<y<d$? – Hello there May 22 at 14:19
• The common regions are $1\le u \le4$ and $0\le v\le1$ – Ak19 May 22 at 14:25