Double Integral $\int\limits_0^1\int\limits_0^{1-x}(y-2x)^{2}\sqrt{x+y}\,dy\,dx$

Question

Good night, i have a serious problem changing the integration limits, i read two books but i don't understand, i put an example... $$\int_0^1\int_0^{1-x}(y-2x)^{2}\sqrt{x+y}\,dy\,dx$$ I use the change: $\begin{cases} u=x+y\\ v=y-2x \end{cases}$

Then: $\begin{cases} x=\frac{u}{3}-\frac{v}{3}\\ y=\frac{2u}{3}-\frac{v}{3} \end{cases}$

And evaluating: $\begin{cases} y+x=1\rightarrow u=1\\ y=0\rightarrow2u=-v\\ x=0\rightarrow u=v\\ x=1\rightarrow v=-3+u \end{cases}$

And the new integral: $$\int_{-3+u}^{-2u}\int_v^1\frac13v^2u^{1/2}\,du\,dv$$

But, this not the answer, what am I doing wrong?

Answer 1

Your expression for $y$ in terms of $v$ are not correct. The expressions should be

$$x=\frac u2-\frac v3, \qquad y=\frac{2u}3+\frac v3$$

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The original, source integral has the outer bounds $$0\le x\le 1$$ and the inner bounds $$0\le y\le 1-x$$

Substituting the correct expressions for $x$ and $y$ give $$0\le\frac u3-\frac v3\le 1, \qquad 0\le\frac{2u}3+\frac v3\le 1-\left(\frac u3-\frac v3\right)$$

Multiplying by three gives $$0\le u-v\le 3, \qquad 0\le 2u+v \le 3-u+v$$

You should graph those inequalities in the $u$-$v$ plane to see what kind of region you get.

You see that is a triangle. Now express overall limits on $u$, then limits on $v$ based on the value of $u$.

Answer 2

The original region of integration is a triangle with vertices $(0,0),\,(0,1)$ and $(1,0)$.

The linear transformation $L(x,y)=(x+y,y-2x)=(u,v)$ will move straight lines to straight lines, so the transformed region will also be a triangle, but its vertices will be $L(0,0)=(0,0),\,L(0,1)=(1,1)$ and $L(1,0)=(1,-2)$.

Draw the triangle of the new region of integration and you will see that the outer limits must go from $u=0$ to $u=1$ and the inner limits must go from the line $v=-2u$ to the line $v=u$.

In other words, it helps to draw the original region of integration and then draw the transformed region. Otherwise you are just manipulating formulas without any understanding of what is going on.