Chap8_Sec1

Chap8_Sec1 - 8 FURTHER APPLICATIONS OF INTEGRATION OF...

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FURTHER APPLICATIONS FURTHER APPLICATIONS OF INTEGRATION OF INTEGRATION 8

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FURTHER APPLICATIONS OF INTEGRATION In chapter 6, we looked at some applications of integrals: Areas Volumes Work Average values
FURTHER APPLICATIONS OF INTEGRATION Here, we explore: Some of the many other geometric applications of integration—such as the length of a curve and the area of a surface Quantities of interest in physics, engineering, biology, economics, and statistics

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FURTHER APPLICATIONS OF INTEGRATION For instance, we will investigate: Center of gravity of a plate Force exerted by water pressure on a dam Flow of blood from the human heart Average time spent on hold during a customer support telephone call
8.1 Arc Length In this section, we will learn about: Arc length and its function. FURTHER APPLICATIONS OF INTEGRATION

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ARC LENGTH What do we mean by the length of a curve?
ARC LENGTH We might think of fitting a piece of string to the curve here and then measuring the string against a ruler.

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ARC LENGTH However, that might be difficult to do with much accuracy if we have a complicated curve.
ARC LENGTH We need a precise definition for the length of an arc of a curve—in the same spirit as the definitions we developed for the concepts of area and volume.

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POLYGON If the curve is a polygon, we can easily find its length. We just add the lengths of the line segments that form the polygon. We can use the distance formula to find the distance between the endpoints of each segment.
ARC LENGTH We are going to define the length of a general curve in the following way. First, we approximate it by a polygon. Then, we take a limit as the number of segments of the polygon is increased.

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ARC LENGTH This process is familiar for the case of a circle, where the circumference is the limit of lengths of inscribed polygons.
ARC LENGTH Now, suppose that a curve C is defined by the equation y = f ( x ), where f is continuous and a x b .

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ARC LENGTH We obtain a polygonal approximation to C by dividing the interval [ a, b ] into n subintervals with endpoints x 0 , x 1 , . . . , x n and equal width Δx.
ARC LENGTH If y i = f ( x i ), then the point P i ( x i , y i ) lies on C and the polygon with vertices P o , P 1 , …, P n , is an approximation to C .

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ARC LENGTH The length L of C is approximately the length of this polygon and the approximation gets better as we let n increase, as in the next figure.
ARC LENGTH Here, the arc of the curve between P i– 1 and P i has been magnified and approximations with successively smaller values of Δx are shown.

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This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.

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Chap8_Sec1 - 8 FURTHER APPLICATIONS OF INTEGRATION OF...

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