The infinite sums arising in the calculation of areas occur so frequently, they have been given their own notation. The summation (greek Sigma) is replaced by an elongated S (integral sign), and the intervals (greek)Delta x are replace by dx.

In the special case where f(x) > 0, the integral of f from a to b is precisely the area underneath the graph of f from a to b.

If we choose the approximating points x_i^* appropriately, we arrive at special numerical integration formulas for example,

• Left point rule: choose x_i^* = x_i-1
• Midpoint rule: choose x_i^* = 1/2*(x_i-1+x_i)
• Right point rule: choose x_i^* = x_i

Ultimately, because summation is linear, definite integration is a linear operation.

Integration also preserves certain inequality relations (see p. 418)

Last modified Wed Nov 6 23:04:49 CST 1996