Introduction

Managing the Klamath River

The Klamath River starts in the eastern lava plateaus of Oregon, passesthrough a farming region of that state, and then crosses northern California.Because it is the largest river in the region, runs through a variety ofterrain, and has different land uses, it is important in a number of ways.Historically, the Klamath supported large salmon runs. It is used toirrigate agricultural lands, and to generate electric power for the region.Downstream, it runs through federal wild lands, and is used for fishing,rafting, and kayaking.

If a river is to be well-managed for such a variety of uses, its flow must beunderstood. For that reason, the U.S. Geological Survey (USGS) maintains anumber of gauges along the Klamath. These typically measure the depthof the river, which can then be expressed as a rate of flow in cubic feetper second (ft\(^{\scriptsize\hbox{3}}\!\)/s). In order to understand the river flowfully, we must be able to find the total amount of water that flows downthe river over any period of time.

CHAPTER 5 PROJECT

The Chapter Project on page 403 examines ways to obtain the total flow over any period of time from data thatprovide flow rates.

We began our study of calculus by asking two questions from geometry. Thefirst, the tangent problem, “What is the slope of the tangent line to thegraph of a function?” led to the derivative of a function.

The second question was the area problem: Given a function \(f\), defined andnonnegative on a closed interval \([a,b]\), what is the area enclosed by thegraph of \(f\), the \(x\)-axis, and the vertical lines \(x = a\) and\(x = b\)? Figure 5.1 illustrates this area.

Figure 5.1: A is the area enclosed by the graph of \(f\), the \(x\)-axis, and the lines \(x\,=\,a\) and \(x\,=\,b\).

The first two sections of Chapter 5 show how the concept of the integralevolves from the area problem. At first glance, the area problem and thetangent problem look quite dissimilar. However, much of calculus is builton a surprising relationship between the two problems and their associatedconcepts. This relationship is the basis for the Fundamental Theoremof Calculus, discussed in Section 5.5.

344