CHAPTER 2 SOME APPLICATIONS OF INTEGRATION Chapter 2 is a bunch of somewhat unrelated topics each of which illustrates an example of how you can give a meaning to an integral Note I have only written down what I think are the most important definitions and theorems in here I expect you to know roughly how the proofs work and you should read those in the book Moreover I will not restrict myself to ask just about these items on the midterm final the full subject of those exams will be determined by a list of sections in the book 1 Physical interpretations Rb If f x describes the rate of change of some physical system then a f x dx describes how much the system has changed if you R moved from a to b For example if f x is your speed as a function of time then f x dx gives the distance travelled during a given period of time Also if f x describes the change in elevation per R distance for a road then f x dx integral over distance gives the total difference in height over the entire trip In the book this is illustrated by the concept of work in Sections 2 14 2 15 2 Areas Areas are discussed in Sections 2 2 2 4 We already know that the integral of a positive function is the area of its ordinate set Slightly generalizing we immediately find Theorem 2 1 Let f and g be integrable functions and let f g on a b then the area of the region S x y x a b f x y g x is given by Z b g x f x dx a As an important example we find that the area of a disc of radius r is given by Z Z r p p 2 2 2 r x dx r 11 1 x2 dx r2 r where the last equation defines for us By looking at the graph of a function from the side we can determine the integral of functions like x1 p for p N on intervals a b with a 0 It is equal to Z b b n 1 n a n 1 n x1 p n 1 n a 3 Trigonometric functions Another example treated in Sections 2 5 2 8 is that we can define and subsequently integrate the trigonometric functions The trigonometric functions are probably the most important non polynomial functions around so you should be 1 2 CHAPTER 2 SOME APPLICATIONS OF INTEGRATION able to work with them with relative ease in particular you should know or be able to quickly recover the doubling and addition formulas The definition is given in Section 2 7 see picture there where if we define A 1 0 as the far right point of the circle around M 0 0 with radius 1 and have a point P with AM P x we have x coordinate of P equal to cos x and the y coordinate of P equal to sin x With some geometry we can now determine the properties as listed in Theorem 2 3 on page 96 and the fundamental properties on page 95 Together these properties allow us to determine Theorem 3 1 Let a b R then Z b cos x dx sin b sin a a Z b sin x dx cos b cos a a 4 Polar coordinates In section 2 9 2 11 it is described how to use polar coordinates to map out a region See pictures there Definition 4 1 The polar coordinates of a point x y are given as r for which x r cos y r sin This means r is the distance from x y to the origin and the angle 1 0 0 0 x y Now we have the important theorem Theorem 4 2 Given f a b R 0 with a b 0 2 Let using r as the polar coordinates of x y the radial set of f be given by S x y 0 r f a b 2 Then if f is integrable the area of S is given by Z 1 b 2 f x dx 2 a 5 Volume Sections 2 12 and 2 13 deal with volume As with area we can t assign a volume to every solid Thus we have to restrict A special class of solids are the Cavalieri solids Definition 5 1 A Cavalieri solid S is a solid for which there exists a line L such that the intersection of S with any plane orthogonal to L is measurable The cross sectional area function aS is given as the areas of those cross sections so it depends on the points in L We can determine the volume of a Cavalieri solid if this function aS is integrable Theorem 5 2 If S is a Cavalieris solid with integrable cross sectional area funcRb tion aS with aS 0 outside a b then the volume of S is given by a aS x dx As an important special case we have Definition 5 3 A solid of revolution associated to function f a b R is the set S x y z a x b y 2 z 2 f x with volume CHAPTER 2 SOME APPLICATIONS OF INTEGRATION 3 Theorem 5 4 Let S be the solid of revolution associated to f a b R Suppose Rb f 2 is integrable then the volume of S is given by a f x 2 dx 6 Averages Sections 2 16 2 17 deal with averages Definition 6 1 Let f a b R be integrable then we define the average of f over the interval a b as Z b 1 f x dx A f b a a Suppose w a b R is an integrable function the weight and wf is integrable over a b then we define the weighted average of f as Rb f x w x dx Aw f a R b w x dx a It should be observed that the ordinary Pnaverage is just the weighted average with constant weight Also the average n1 k 1 ak of a set of numbers a1 a2 an equals in this case the average of the step function on 0 n defined by f x ak for x k 1 k and arbitrarily on the integers 7 Indefinite integrals Sections 2 18 and 2 19 deal with indefinite integrals Indefinite integrals are defined as Definition 7 1 Let f a b R be an integrable function then an indefinite integral of f is given by a function F a b R such that Z d f x dx F d F c c for any c d with c d a b The example of an indefinite integral is Z x A x f x dx a and all other indefinite integrals differ from A by just a constant independent of x We are now interested in what properties we can derive for the indefinite integral given properties for f Theorem 7 2 Suppose f …