Average value for multiple integrals

Question

If there is a function $f(x,y)$ and we want to find the average value over a region $R$ defined by $0<x<1$ and $0<y<x$, how is that computed? I know that it would be something like this: $$\frac{1}{A(R)}\int_0^1\!\!\!\int_0^{x}f(x,y)\,dydx$$ But I don't know how to find $A(R)$, which is the area of the region $R$, numerically. How can I end up with a numerical value?

Answer 1

The hard way is to compute $A(R) = \displaystyle\iint\limits_{R}\,dx\,dy = \int_{0}^{1}\int_{0}^{x}\,dy\,dx$.

The easy way is to note that $R$ is a right triangle with base length $1$ and height $1$.