1.)$f(x)$ is continuous in $0\le x\le a$, prove that :
$$\iint_Df(x+y) dx dy= \int_{0}^{a} x f(x) dx$$ where $D=\{(x,y)|x \geq 0, y \geq 0, x+y \le a\}$
i try to change variable to u and v coordinate
$x+y=v$ and $u=x$
According to the domain if i draw in xy plane , it is a triangle in first quadrant with equation $y=-x+a$ so D = $0 \le x \le a$ & $0 \le x+y \le a$
but my book said that the domain is $0 \le x \le a$ , $x \le x+y \le a$, how can i get this domain?
and then find jacobian from $u=x$ and $v=x+y$.
$$\det\bigg(\begin{bmatrix} 1& 0\\
-1 & 1
\end{bmatrix}\bigg) =1$$
is it correct? and for this problem is it necessary to change variable to $uv$ plane? thanks !
Your domain $D$ is a triangular region in the first quadrant bounded by the three lines $x=0$, $y=0$ and $x+y=a$. You can view this as a region bounded on the left and right by two vertical lines $x=0$ and $x=a$, hence $$0\leq x\leq a.$$ It is bounded below and above by the lines $y=0$ and $x+y=a$ or $y=a-x$. So $0\leq y\leq a-x$ which, after adding $x$ everywhere, is the same thing as $$x\leq x+y\leq a.$$