3
Applications of Definite Integrals to rates, velocities, and densities
Velocity, acceleration, and displacement 3.1
Two cars, labeled 1 and 2 start side by side and accelerate from rest. Figure 1 shows a graph of their velocity functions, with t measure

8
8.1
Differential Equations
Use separation of variables to solve the following differential equations with given initial conditions. (a) (b)
dy dt dy dt
= -2ty, y(0) = 10
= y(1 - y), y(0) = 0.5, (Hint:
1 y(y-1)
=
1 y-1
1 - y ).
Solution (a) (b)
dy y
= -2

9
Infinite Series, Improper Integrals, and Taylor Series
Sequences and series 9.1
Determine which of the following sequences converge or diverge (a) cfw_en (b) cfw_2-n (c) cfw_ne-2n 2 (d) cfw_ n n (e) cfw_ 2 (f) cfw_ln(n)
Solution (a) limn en = , div

7
Probability
Discrete Probability
7.1
Multiple Events and Combined Probabilities 1
Determine the probability of each of the following events assuming that the die has equal probability of landing on each one of the six sides marked by 1 to 6 dots and tha

6
6.1
Mass distribution and Center of Mass
In a class of 20 students writing a test worth 10 points, 5 students scored 6 points, 5 scored 8 points, and 10 scored 9 points. Find the average score achieved by this class on the test. Solution The average gra

5
5.1
Techniques of Anti-differentiation
Differential Notation 1
Calculate the differential of the following functions by using the definition dy = y (x)dx. Express the result in terms of the product between y (x) and the differential of x, dx. For exampl

4
More Applications of Definite Integrals: Volumes, arclength and other matters
Volumes of surfaces of revolution 4.1
Find the volume of a cone whose height h is equal to its base radius r, by using the disc method. We will place the cone on its side, as

2
Areas and the Fundamental Theorem of Calculus
Area Under the Curves
2.1
Estimate the area under the graph of f (x) = x2 + 2 from x = -1 to x = 2 in each of the following ways, and sketch the graph and the rectangles in each case. (a) By using three rect

1
1.1
Summation: Adding up the pieces
Answer the following questions: (a)What is the value of the fifth term of the sum S = (b)How many terms are there in total in the sum S = (c)Write out the terms in (d)Write out the terms in
5 n=1 4 n=0 20
(5 + 3k)/k?

Chapter 10
Innite series, improper
integrals, and Taylor
series
10.1 Introduction
This chapter has several important and challenging goals. The rst of these is to understand how concepts that were discussed for nite series and integrals can be meaningfull

Chapter 9
Differential Equations
9.1
Introduction
A differential equation is a relationship between some (unknown) function and one of its
derivatives. Examples of differential equations were encountered in an earlier calculus
course in the context of pop

Chapter 8
Continuous probability
distributions
8.1
Introduction
In Chapter 7, we explored the concepts of probability in a discrete setting, where outcomes
of an experiment can take on only one of a nite set of values. Here we extend these
ideas to contin

Chapter 7
Discrete probability and
the laws of chance
7.1
Introduction
In this chapter we lay the groundwork for calculations and rules governing simple discrete
probabilities24. Such skills are essential in understanding problems related to random proces

Chapter 6
Techniques of
Integration
In this chapter, we expand our repertoire for antiderivatives beyond the elementary functions discussed so far. A review of the table of elementary antiderivatives (found in Chapter 3) will be useful. Here we will discu

Chapter 5
Applications of the
denite integral to
calculating volume,
mass, and length
5.1
Introduction
In this chapter, we consider applications of the denite integral to calculating geometric
quantities such as volumes of geometric solids, masses, center

Chapter 4
Applications of the
denite integral to
velocities and rates
4.1
Introduction
In this chapter, we encounter a number of applications of the denite integral to practical
problems. We will discuss the connection between acceleration, velocity and d

Chapter 3
The Fundamental
Theorem of Calculus
In this chapter we will formulate one of the most important results of calculus, the Fundamental Theorem. This result will link together the notions of an integral and a derivative.
Using this result will allo

Chapter 2
Areas
2.1
Areas in the plane
A long-standing problem of integral calculus is how to compute the area of a region in
the plane. This type of geometric problem formed part of the original motivation for the
development of calculus techniques, and