Consider the region $R$ which is the region bounded between the four points
$$
(0,b) \ , \ \ (a,a+b)\ , \ \ (a,c) \ , \ \ (0,c)
$$
where $0<a<b$ and $a+b<c$, which looks like:
For a function $f(x,y)$ I would like to write the integral
$$
\iint_R f(x,y) dx dy
$$
as a single integral.
I can write this as a sum of two terms (over a triangular and rectangular region):
$$
\iint_R f(x,y) dx dy = \overbrace{\int_0^a dx \int_{x+b}^{a+b} dy\ f(x,y) }^{\text{triangle between }(0,b)\ \& \ (0,a+b) \ \& \ (a,a+b) }\ \ + \ \ \overbrace{\int_0^a dx \int_{a+b}^{c} dy\ f(x,y)}^{\text{rectangle between }(0,a+b)\ \& \ (a,a+b) \ \& \ (a,c) \ \& \ (0,c) }
$$
QUESTION: Is there a way to write this integral without splitting it apart into different regions?
I am tempted to write the above as
$$
\iint_R f(x,y) dx dy = \int_0^a dx \int_{x+b}^{c} dy\ f(x,y)
$$
but I don't think this is correct.
Best Answer
What you are tempted to write is correct. Please note the region is bound between vertical lines $x = 0$ and $x = a$ and horizontal lines $y = c ~ $ and $ ~ y = x + b$.
(Based on your diagram, assuming $c \gt a + b \gt b$).
So the double integral can indeed be set up as,
$ \displaystyle \int_0^a\int_{x+b}^c f(x, y) ~ dy ~ dx$