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 \(\dx=(b-a)/n\) 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
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
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 Idea7.0.1.Application of Definite Integrals Strategy.
Let a quantity be given whose value \(Q\) is to be computed.
Divide the quantity into \(n\) smaller “subquantities” of value \(Q_i\text{.}\)
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.
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.
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.