How do you calculate the area of an irregular region such as one partially bounded by rivers?

Answer 1

In real-world situations, the best that you can do is approximate using numerical methods. For instance, you could take an aerial photograph of the region, overlay a grid, and then use methods such as "The trapezoid rule" or "Simpson's rule". Any calculus text should cover how to use these and you can also find a description of these methods at:

http://en.wikipedia.org/wiki/Trapezoidal_rule

http://en.wikipedia.org/wiki/Simpson%27s_rule

If, however, it is a math problem and you are given functions of the curves (or rivers), then you just integrate the difference (in absolute value) of the two functions. For instance, if we have the curves f and g on [a,b], and f is greater than or equal to g on the interval, then we just integrate f-g over [a,b].