6 Applications of Integration

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


Let Δx=(b-a)/n denote the length of the subintervals, and let ci be any x-value in the i th subinterval. Definition 5.3.1 states that the sum


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


Theorem 5.3.2 connects limits of Riemann Sums to definite integrals:


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.

  1. (a)

    Divide the quantity into n smaller “subquantities” of value Qi.

  2. (b)

    Identify a variable x and function f(x) such that each subquantity can be approximated with the product f(ci)Δx, where Δx represents a small change in x. Thus Qif(ci)Δx.

  3. (c)

    Recognize that Qi=1nQi=i=1nf(ci)Δx, which is a Riemann Sum.

  4. (d)

    Taking the appropriate limit gives Q=abf(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.

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