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

Let $\dx=(b-a)/n$ denote the length of the subintervals, and let $c_i$ be any $x$-value in the $i^\text{ th }$ subinterval. Definition 5.3.12 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. \end{equation*}

Theorem 5.3.20 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. \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 Idea7.0.1Application 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. 5.4.