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
$$ |
Let $\mathrm{\Delta}x=(b-a)/n$ denote the length of the subintervals, and let ${c}_{i}$ be any $x$-value in the ${i}^{\text{th}}$ subinterval. Definition 5.3.1 states that the sum
$$\sum _{i=1}^{n}f({c}_{i})\mathrm{\Delta}x$$ |
is a Riemann Sum. Riemann Sums are often used to approximate some quantity (area, volume, work, pressure, etc.). The approximation becomes exact by taking the limit
$$\underset{n\to \mathrm{\infty}}{lim}\sum _{i=1}^{n}f({c}_{i})\mathrm{\Delta}x.$$ |
Theorem 5.3.2 connects limits of Riemann Sums to definite integrals:
$$\underset{n\to \mathrm{\infty}}{lim}\sum _{i=1}^{n}f({c}_{i})\mathrm{\Delta}x={\int}_{a}^{b}f(x)\mathit{d}x.$$ |
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.
Let a quantity be given whose value $Q$ is to be computed.
Divide the quantity into $n$ smaller “subquantities” of value ${Q}_{i}$.
Identify a variable $x$ and function $f(x)$ such that each subquantity can be approximated with the product $f({c}_{i})\mathrm{\Delta}x$, where $\mathrm{\Delta}x$ represents a small change in $x$. Thus ${Q}_{i}\approx f({c}_{i})\mathrm{\Delta}x$.
Recognize that $Q\approx {\displaystyle \sum _{i=1}^{n}}{Q}_{i}={\displaystyle \sum _{i=1}^{n}}f({c}_{i})\mathrm{\Delta}x$, which is a Riemann Sum.
Taking the appropriate limit gives $Q={\displaystyle {\int}_{a}^{b}}f(x)\mathit{d}x$
This Key Idea will make more sense after we have had a chance to use it several times. We begin with Area Between Curves.