Answer to Question #5792 in Geometry for maria prettymaths

Question

How to derive Area formula.

Answer 1

Let's derive area formula for plane objects.
A definite integral of a function can be represented as the signed area of the region bounded by its graph: S = int(f(x))dx from a to b.
We can bound any figure on R² by two real functions f1(x) and f2(x) with x from some interval [a,b]. Then the area of the figure bounded by this functions is
S = int(f1(x))dx from a to b - int(f2(x))dx from a to b = int(f1(x)-f2(x))dx from a to b.
Using obtained formula we can calculate area of any bouded region on R². Let's do it for the [0;1]x[0;2] rectangle for show. Here f1(x) = 2, xє[0;1] and f2(x) = 0, xє[0;1], so
S = int(f1(x)-f2(x))dx from 0 to 1 = int(2-0)dx from 0 to 1 = 2*1 = 2.