24.2 The Average Value of a Function: An Application of the DeÞnite Integral 781
6. The amount of a certain chemical in a mixture varies with time. If ! 5 is the number of grams of the chemical at time , what is the average number of grams of the chemical in the mixture on the time interval [0, 1]?
7. The velocity of an object is given by 3 sin".
(a) What is the objectÕs speed as a function of time?
(b) What is the objectÕs net displacement from 0 to 2? (c) How far has the object traveled from 0 to 2?
(d) What is the objectÕs average velocity on [0, 2]? (e) What is the objectÕs average speed on [0, 2]?
8. The graphs of functions , !, , and are given below. Let I denote the average value of on [0, 4].
Let II denote the average value of ! on [0, 4]. Let III denote the average value of on [0, 4]. Let IV denote the average value of on [0, 4].
Put I, II, III, and IV, in ascending order, with ÒÓ or ÒÓ signs between them as appropriate.
f g h k
(2, 5) (2, 5) (2, 5)
0 4 0 4 0 4 0 4
9. A bicycle speedometer will give the average velocity of a bicyclist over the time period the bicycle is moving. By pressing a button the bicylist can reset the average velocity counter. Suppose a long-distance cyclist has averaged 14 miles per hour for the Þrst two hours of her trip. She resets the average velocity counter. For the next four hours her average velocity is 18 miles per hour.
(a) What is the cyclistÕs average velocity for the six-hour trip? (b) How far has she traveled?
10. It takes a bicyclist 8 minutes to ride 1 mile uphill and then 2 minutes to ride 1 mile downhill. Explain how we know that the cyclistÕs average velocity for the hill is 12 miles per hour.
11. Thetemperatureofahotplateofradius5inchesvarieswiththedistancefromthecenter of the plate. For the area within 2 inches of the center the average temperature is 100 degrees. For the area between 2 and 5 inches from the center the average temperature is 80 degrees. What is the average temperature of the plate?
12. Find the average value of sin3 on [0, 2"]. Explain your reasoning.
782 CHAPTER 24 The Fundamental Theorem of Calculus
13. The graph below shows the birthrate, ’, and the death rate, (, for a population of Þsh in a lake. is measured in years, and 0 represents 1960. The population of Þsh in 1960 was 4500. (Assume that births and deaths are the only factors that affect the populationÑno Þshing, no immigration, etc.)
0 4 8 12 16 20 24 28 32 36 t
(a) Write an expression for the total number of births between 1960 and 1996. (b) Write an expression for the average death rate between 1980 and 1990.
(c) Write an expression for the Þsh population in 1996. (d) Approximate the year the population was greatest.
(e) Was the 1996 population greater or less than 4500? Explain.
P A R T IX
Applications and Computation of the Integral
C H A P T E R
Finding Antiderivatives An Introduction to Indenite Integration
25.1 A LIST OF BASIC ANTIDERIVATIVES
InordertoapplytheFundamentalTheoremofCalculusweneedtobeabletoÞndantideriva-tives. This process can be challenging. Therefore, it is useful to have at our disposal a list of functions we can readily antidifferentiate. We obtain this list by thinking about functions we can readily differentiate and working backward.
D e f i n i t i o n
Thesymbol standsfortheentirefamilyoffunctionsthatareantiderivatives of . is called the indeÞnite integral of . is the integrand.
Recall that if is an antiderivative of , then is also an antiderivative of for any constant .
784 CHAPTER 25 Finding AntiderivativesÑAn Introduction to IndeÞnite Integration
Any two antiderivatives of differ by an additive constant, so every antiderivative of can
be written in the form . We know that
sin cos ; therefore
Equivalently, we can write
cos sin .
sin cos ;
therefore cos sin .
) sin )cos ); therefore cos ) )sin ).
Thevariables,,and)aresometimesreferredtoasÒdummyvariables,Ómeaningthat the statements weÕve made are equivalent, regardless of the variable we use. cos sin , cos sin , and cos ) )sin ) all say the same thing.
arcsin 1 1
arccos 1 1
Corresponding Antiderivatives 1 for 1
1 ln (An explanation of the need for will follow.)
cos sin sin cos cos sin , so [cos ] sin
1 arcsin 1
1 arccos 1
The Ò,Ó in each of these indeÞnite integrals is necessary; without the ÒÓ we have given only one antiderivative as opposed to the entire family of antiderivatives. Fromagraphicalperspective,theÒÓindicatesthatallantiderivativesofthefunction
are vertical translates of one another.
Why do we need the absolute value bars in 1 ln ?
We are looking for an antiderivative of 1.
If 0 and ln , then 1. Therefore, ln is an antiderivative of 1 for 0.
If 0, then ln is undeÞned. However, if we let ln, then 1 . Therefore, ln is an antiderivative of for 0.
25.1 A List of Basic Antiderivatives 785
ln for 0 ln for 0
Therefore, we can use ln as an antiderivative of 1 for all 0.
EXERCISE 25.1 On the one hand, 1 arccos ; on the other hand, 1 1 1
1 arcsin . Therefore, arccos and arcsin must differ by a constant.
In addition to the list of speciÞc antiderivatives given in the table on page 784, we can deduce principles for integration of the sum of functions and integration of a constant multiple of a function from the corresponding principles for differentiation.
General Differentiation Rules
Corresponding General Integration Rules
The antiderivatives listed in the table on page 784 together with the two general integration rules above, enable us to evaluate a wide variety of deÞnite integrals. Learning how to undo the Chain Rule will allow us to evaluate an even broader range of deÞnite and indeÞnite integrals. We will do this, using a technique called substitution, in this chapter. We can further expand the type of functions we can integrate by using the Product Rule in reverse and arriving at a technique called integration by parts. This will be taken up in Section 29.1. Intheproblemsbelowweapplygeneralintegrationrulesincombinationwiththebasic antiderivativesdisplayedinthetableonpage784.Ourstrategyistomanipulatetheintegrand algebraically so that it can be expressed as a sum. Then the integral can be pulled apart into
the sum of simpler integrals.
EXAMPLE 25.1 Integrate the following. (Try these on your own Þrst; then read the solutions.) (a) 1 22 (b) 7 (c) 2 3
SOLUTIONS (a) 1 2
4 22 1
Note that thereÕs no need to
use three separate constants.