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# Integrals

**Overview**

1. Definition of an integral
2. Evaluating integrals using anti-derivatives
3. The fundamental theorem of calculus

**Definition of an integral**

{% hint style="info" %}
The integral is a mathematical concept used to determine the area under a curve
{% endhint %}

The integral is denoted by the <mark style="color:red;">symbol ∫</mark> 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)
$$