3️Integrals

Overview

1. Definition of an integral

2. Evaluating integrals using anti-derivatives

3. The fundamental theorem of calculus

Definition of an integral

The integral is a mathematical concept used to determine the area under a curve

The integral is denoted by the symbol ∫ and has two limits of integration (the lower and upper limits).

Evaluating integrals using anti-derivatives

Antiderivatives (or indefinite integrals) are used to evaluate integrals.

The antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x). If F(x) is an antiderivative of f(x), then we can find the definite integral of f(x) over the interval [a, b] using the following formula:

$$∫[a,b] f(x) dx = F(b) - F(a)$$

The fundamental theorem of calculus

The fundamental theorem of calculus establishes the relationship between differentiation and integration. It states that if f(x) is a continuous function on the interval [a, b], and F(x) is an antiderivative of f(x), then:

$$∫[a,b] f(x) dx = F(b) - F(a)$$