Numerical Integration

Abstract

The integral is the area under a curve, to be more precise

$$\displaystyle{I(f):=\int _{ a}^{b}f(x)\,\mbox{ d}x}$$

is the area under the real function y = f(x) (and between the straight lines y = 0, x = a and x = b > a, resulting in a positive value for f(x) > 0 and a negative value for f(x) < 0). If f is a function of two variables, then the double integral

$$\displaystyle{I(f):=\int _{ a}^{b}\int _{ c}^{d}f(x,y)\,\mbox{ d}x\,\mbox{ d}y}$$

is the volume under the surface z = f(x, y) and over the rectangle a ≤ x ≤ b, c ≤ y ≤ d of the x, y-plane.