APEX Applications of Integration

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

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a = x_{0} < x_{1} < \dots < x_{n} < x_{n} = b .

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Let

Δ x = (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

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\sum_{i = 1}^{n} f (c_{i}) Δ 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

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lim_{n \to \infty} \sum_{i = 1}^{n} f (c_{i}) Δ x .

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Theorem 5.3.26 connects limits of Riemann Sums to definite integrals:

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lim_{n \to \infty} \sum_{i = 1}^{n} f (c_{i}) Δ x = \int_{a}^{b} f (x) d x .

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Finally, the Fundamental Theorem of Calculus states how definite integrals can be evaluated using antiderivatives.

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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.

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Key Idea 7.0.1. Application of Definite Integrals Strategy.

Let a quantity be given whose value

Q

is to be computed.

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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 $f (c_{i}) Δ x,$ where $Δ x$ represents a small change in $x .$ Thus $Q_{i} \approx f (c_{i}) Δ x .$ A sample approximation $f (c_{i}) Δ x$ of $Q_{i}$ is called a differential element.
Recognize that $Q = \sum_{i = 1}^{n} Q_{i} \approx \sum_{i = 1}^{n} f (c_{i}) Δ x,$ which is a Riemann Sum.
Taking the appropriate limit gives $Q = \int_{a}^{b} f (x) d x$

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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.

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