#Definite Integrals Study Guide

#Table of Contents

A definite integral is written as:

$\int_{a}^{b} f(x) , dx$

Integrand ( $f(x)$ ): The function being integrated.
Integration Variable ( $dx$ ): Indicates the variable of integration.
Limits of Integration ( $a$ and $b$ ): The bounds between which the function is integrated.
- The function is integrated 'from $a$ to $b$ '.
- Typically, $a \leq b$ , but $a > b$ is also valid.

Key Concept

Key Concept: A definite integral represents the area under the curve of a function $f(x)$ from $x = a$ to $x = b$ .

**Example**: If

f(x) \geq 0

on the interval

[a, b]

, the definite integral

\int\_{a}^{b} f(x) \, dx

gives the area between the function

f(x)

and the

x

-axis, from

x = a

to

x = b

.

A definite integral outputs a number based on $f(x)$ and the values of $a$ and $b$ .
If $f(x)$ is a rate of change function, the definite integral represents the accumulation of change from $x = a$ to $x = b$ .

**Note**: The definite integral can also be interpreted as the signed area under the curve, where areas above the

x

-axis are positive and those below are negative.

The value of a def...