17.1.1 Definitions and Formulas

Indefinite Integral or antiderivative

F is an antiderivative function of f if F′(x) = f(x). The antiderivative is also called the indefinite integral of f and is denoted by ∫ f(x)dx + C , where C is a constant.

Constant of integration

If F(x) is an anitderivative function of f, then F(x) + C, where C is any real number, is likewise an anitderivative function of f since (F(x) + C) = F(x) + 0 = f(x). C is called the constant of integration for the indefinite integral.

Definite integral and the limit of a Riemann sum

The definite integral from x = a to x = b is the limit of the Riemann sum lim _n→∞∑ _i=0^n-1c′(x _i)Δx as n →∞, where Δx = . The definite integral is denoted by ∫ _a^bf(x)dx.

Lower limit and upper limit

a is called the lower limit of the integral ∫ _a^bf(x)dx and b is called the upper limit.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that ∫ _a^bf(x)dx = F(b) - F(a), where F is an antiderivative of f.

Area Under a Curve

If f(x) is positive from a to b, then the definite integral ∫ _a^bf(x)dx computes the area under f(x) and above the x-axis from a to b.

Numerical Integration

Numerical integration approximates the definite integral

by ∑ _i=0^N-1f(x _i)Δx, where N is a very large number. There are several different methods of numerical integration but this text uses the method of rectangles for simplicity and ease of discussion.