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APEX Calculus: for University of Lethbridge

Chapter7Applications 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_0 \lt x_1 \lt \cdots \lt x_n\lt x_{n}=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 Idea7.0.1.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.