6 Applications of Integration 6 Applications of Integration 6.1 Area Between Curves

Chapter Introduction

We begin this chapter with a reminder of a few key concepts from Chapter 5. Let $f$ be a continuous function on $[a,b]$ which is partitioned into $n$ equally spaced subintervals as

a=x_{0}<x_{1}<\cdots<x_{n-1}<x_{n}=b.

Let $\Delta x=(b-a)/n$ denote the length of the subintervals, and let $c_{i}$ be any $x$ -value in the $i^{\text{ th}}$ subinterval. Definition 5.3.1 states that the sum

\sum_{i=1}^{n}f(c_{i})\Delta x

is a Riemann Sum. Riemann Sums are often used to approximate some quantity (area, volume, work, pressure, etc.). The approximation becomes exact by taking the limit

\lim_{n\to\infty}\sum_{i=1}^{n}f(c_{i})\Delta x.

Theorem 5.3.2 connects limits of Riemann Sums to definite integrals:

\lim_{n\to\infty}\sum_{i=1}^{n}f(c_{i})\Delta x=\int_{a}^{b}f(x)\ dx.

Finally, the Fundamental Theorem of Calculus states how definite integrals can be evaluated using antiderivatives.

This chapter employs the following technique to a variety of applications. Suppose the value $Q$ of a quantity is to be calculated. We first approximate the value of $Q$ using a Riemann Sum, then find the exact value via a definite integral. We spell out this technique in the following Key Idea.

Key Idea 6.0.1 Application of Definite Integrals Strategy

Let a quantity be given whose value $Q$ is to be computed.

(a)

Divide the quantity into $n$ smaller “subquantities” of value $Q_{i}$ .
(b)

Identify a variable $x$ and function $f(x)$ such that each subquantity can be approximated with the product $f(c_{i})\Delta x$ , where $\Delta x$ represents a small change in $x$ . Thus $Q_{i}\approx f(c_{i})\Delta x$ .
(c)

Recognize that $\displaystyle Q\approx\sum_{i=1}^{n}Q_{i}=\sum_{i=1}^{n}f(c_{i})\Delta x$ , which is a Riemann Sum.
(d)

Taking the appropriate limit gives $\displaystyle Q=\int_{a}^{b}f(x)\ dx$

This Key Idea will make more sense after we have had a chance to use it several times. We begin with Area Between Curves.