Determining Limits of Integration - Multiple Integrals

Question

I was wondering whether someone could refresh my memory of something I think I remember learning when taking Calculus II or some similar class. Say, for example, that one has a function of a couple of variables, with respect to each of which one intends to integrate. Suppose there are also multiple conditions for the variables to satisfy. What method is used to convert the conditions into something that can be used to construct an integral?

In the event that this is not entirely clear, allow me to provide an example. Suppose one wishes to sum the function $f(x, y)$ with the conditions that $0 < x < 1$, $0 < y < 1$, and $x + y = 1$. If, for example, we solve this last condition for $y$, we find that $y = 1 - x$. Since the value of $(1 - x)$ is nonnegative for all $x \in \left(0, 1\right)$, can we simply integrate as $$\int_{0}^{1}{\int_{0}^{1-x}{f(x,y)} \hspace{5 px} dy} \hspace{5 px} dx?$$

Answer 1

Note that the following integral

$$\int_{0}^{1}{\int_{0}^{1-x}{f(x,y)} \hspace{5 px} dy} \hspace{5 px} dx$$

corresponds to the triangular region

• $0\le x\le 1$
• $0\le y\le 1-x$

Answer 2

The integral that you wrote has conditions $0\le x\le1$ and $0\le y\le 1-x$.

If you put the condition $x+y=1$, then the integral is one dimensional, and can be written as $\int_0^1f(x,1-x)dx$, or using delta functions