Introduction

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The Cooling Towers at Brayton Point Power Station

The scale of the massive $620 million cooling towers project at the Brayton Point Power Station in Somerset, Massachusetts, is best told by the numbers:

  • Two 497-foot-tall towers, 406 feet across at the base
  • A capacity to pump more than an Olympic-size swimming pool worth of water each minute
  • The use of enough concrete in both towers to pave a sidewalk 250 miles long

For such a huge project, the purpose is simple: Brayton Point must reuse the water it uses to cool the steam from turbines instead of sending it into Mount Hope Bay, pursuant to an order from the Environmental Protection Agency.

CHAPTER 6 PROJECT

The Chapter Project on page 469 investigates the design of cooling towers and determines the quantity and cost of the concrete needed to build one.

When working with definite integrals, we rely on two facts. First, if a function \(f\) is continuous on a closed interval \([a,b]\), then \(f\) is integrable over \([a,b]\). That is, the definite integral of \(f\) from \(a\) to \(b\) exists and is equal to the limit of the Riemann sums. \begin{equation*} \int_{a}^{b}f (x)~{\it dx}=\lim_{n\rightarrow \infty }\sum_{i=1}^{n}f(u_{i})\Delta x=\hbox{a number} \end{equation*}

where \(\Delta x=\dfrac{b-a}{n}\). Second, if a function \(f\) is continuous on a closed interval \([a,b]\) and if \(F\) is any antiderivative of \(f\) on \(( a,b)\), then \[ \int_{a}^{b}f (x)~{\it dx}=F(b)-F(a) \]

We begin this chapter by using a definite integral to solve geometry problems. First we find the area enclosed by the graphs of two or more functions, then we investigate several methods for finding the volume of a solid of revolution, as well as a method for finding the length of a graph. In later sections, we investigate how to use a definite integral and the Fundamental Theorem of Calculus to compute work, to find hydrostatic pressure, and to calculate the centroid of a lamina.

In each application, the words are different, but the melody is the same. The quantity we seek is partitioned into small segments. Each segment is estimated and Riemann sums are obtained. Then we allow the number of segments to grow without bound and express the quantity we seek as a definite integral.