Next Topic - Position, Velocity & Acceleration

#Definite Integrals as Accumulated Change

#Table of Contents

Introduction to Definite Integrals
Net Change in Applied Contexts
Real-World Examples
Exam Tips
Worked Examples
- Example 1: Marginal Cost and Profit
- Example 2: Food Consumption by Rabbits
Glossary
Practice Questions
Summary and Key Takeaways

#Introduction to Definite Integrals

A definite integral represents the accumulation of a rate of change over an interval. It can be used to calculate the net change of a quantity in various applied contexts.

#Net Change in Applied Contexts

Key Concept

Let $f(x)$ represent the rate of change of a function $F(x)$ . This can be expressed as: [ f(x) = \frac{d}{dx}F(x) = F'(x) ]

The net change of $F$ between $x = x_1$ and $x = x_2$ is given by the integral: [ \int_{x_1}^{x_2} f(x) , dx ]

#Understanding the Formula

$F$ over a small interval $\Delta x$
$f(x) , dx$ is the limit of this change as $\Delta x \to 0$
The integral sums these small changes over the interval from $x = x_1$ to $x = x_2$

For a deeper understanding of definite integrals as the limit of a sum, refer to the 'Properties of Definite Integrals' study guide.

#Value of the Function at a Specific Point

The value of $F$ at $x_2$ is: [ F(x_2) = F(x_1) + \int_{x_1}^{x_2} f(x) , dx ]

Similarly, the value of $F$ for any $x$ can be written as: [ F(x) = F(x_0) + \int_{x_0}^{x} f(s) , ds ] where $s$ is a dummy variable used for integration.

#Real-World Applications

Consider a company's **profit** (

P(x)

), **revenue** (

R(x)

), and **cost** (

C(x)

): \[ P(x) = R(x) - C(x) \] where

x

represents the units of merchandise sold.

Marginal profit, marginal revenue, and marginal cost are given by: [ P'(x) = R'(x) - C'(x) ] These represent the rates of change of their respective quantities when $x$ increases by 1.

#Time as an Independent Variable

Often, the independent variable in real-world situations is time. If $f(t)$ is the rate of change of a quantity $F(t)$ ...

Previous Topic - Average Value of a Function Next Topic - Position, Velocity & Acceleration

Definite Integrals in Context

Sarah Miller