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Chapter 7 Applications of Integration

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

\begin{equation*} a=x_1 \lt x_2 \lt \cdots \lt x_n\lt x_{n+1}=b\text{.} \end{equation*}

Let \(\dx=(b-a)/n\) denote the length of the subintervals, and let \(c_i\) be any \(x\)-value in the \(i\)th subinterval. Definition 5.3.17 states that the sum

\begin{equation*} \sum_{i=1}^n f(c_i)\dx \end{equation*}

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

\begin{equation*} \lim_{n\to\infty} \sum_{i=1}^n f(c_i)\dx\text{.} \end{equation*}

Theorem 5.3.26 connects limits of Riemann Sums to definite integrals:

\begin{equation*} \lim_{n\to\infty} \sum_{i=1}^n f(c_i)\dx = \int_a^b f(x)\, dx\text{.} \end{equation*}

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 7.0.1. Application of Definite Integrals Strategy.

Let a quantity be given whose value \(Q\) is to be computed.

  1. Divide the quantity into \(n\) smaller “subquantities” of value \(Q_i\text{.}\)

  2. Identify a variable \(x\) and function \(f(x)\) such that each subquantity can be approximated with the product \(f(c_i)\dx\text{,}\) where \(\dx\) represents a small change in \(x\text{.}\) Thus \(Q_i \approx f(c_i)\dx\text{.}\) A sample approximation \(f(c_i)\dx\) of \(Q_i\) is called a differential element.

  3. Recognize that \(\ds Q= \sum_{i=1}^n Q_i \approx \sum_{i=1}^n f(c_i)\dx\text{,}\) which is a Riemann Sum.

  4. Taking the appropriate limit gives \(\ds Q = \int_a^b f(x)\, dx\)

This Key Idea will make more sense after we have had a chance to use it several times. We begin with Area Between Curves, which we addressed briefly in Section 5.4.