5.10 Improper integrals. A: Infinite interval of integration.
Understand how to calculate an integral from a to
infinity as the limit, as T goes to infinity, of the
integral from a to T if that limit exists.
[Fundamental examples 1/x2 on page 428 and
1/x in Example 1]. Understand why 1/xp
gives a convergent integral from 1 to infinity if p > 1
and a divergent integral (no limit) otherwise [Example 4].
B: Function goes to infinity at a finite value a. Understand how to calculate an integral from a to b as the limit, as t goes to a, of the integral from t to b if that limit exists. Fundamental examples 1/x1/2 [Exercise 23] and 1/x. Understand why 1/xp gives a convergent integral from 0 to 1 if p < 1 and a divergent integral (no limit) otherwise [Exercise 49]. Exercises 5,17.
6.1 Areas between curves. Understand that if f > g on an interval [a,b] then the area between the graphs is the area under f minus the area under g [Example 1]. Know how to solve a "region enclosed" problem: locate the intersection points - these will be the limits of integration [Example 2]. Be able to set up the problem as a y-integral when appropriate [Example 5]. Exercises 7,11.
6.2 Volumes. Understand how slicing reduces the calculation of volume to a calculation of area and an integration [Discussion on pages 454 and 455] and how to implement the calculation [sphere, Example 1]. Know how to set up the integral for the volume of a solid of revolution [Examples 2, 5]. Know how to apply slicing to set up the volume integral for other solids [Example 7 and Example 8]. Be able to use the "cylindrical shell" method when appropriate [Example 9]. Exercises 5, 9, 21, 23, 25, 43.
6.5 Work. Understand that if the force is a constant F, and displaces
its application point from a to b, the work done is
the product W = F (b-a); and that if the force varies with
distance x as F(x) the problem is handled by slicing
the interval [a,b] into infinitesimal subintervals of length
dx from x to x + dx over which the force can
be considered constant. The displacement from x to x + dx
involves an infinitesimal amount of work dW = F(x)dx, and
these infinitesimal amounts of work are summed by the integral: the
W is the integral of dW from a to b.
Be able to to convert problems of this type into integrals.
[Examples 1 and 2 page 478, Exercises 1-6].
A different application of Calculus is to problems where the distance that points are moved varies with the parameter x. In the simplest examples (e.g. emptying a rectangular tank of water over the top) the force is uniform, but points at one end of the problem (the bottom) have to travel farther than those at the other end (the top; here x is the height). The method is to slice the problem perpendicular to the x-direction, and to let dW be work corresponding to the slice at height x and thickness dx. Exercises 7-9, 11, 12.
In more complicated examples, the size of the slice may also vary with x [Example 3 page 479, Exercises 13, 14]. Or the force may also vary with x [Exercise 10].
Be able to apply the "slice and integrate" method to all these problems.
6.7 Probability. Be able to check whether a given function f(x) can be a probability density function [Examples 1,2 page 493] and how to interpret the integral from a to b of f(x)dx as a probability [Exercises 2,3,4ab]. Be able to calculate the average value (= the mean) of the random variable described by a probability density function f(x) [Example 3 page 495, Exercise 4c]. Be able to work with probability density functions given by graphs [Exercise 4] and equations [Example 1]; and in particular with exponentially decreasing probability density functions [Example 2, Examples 3 and 4 page 495, Exercises 6,7].
7.2 Direction fields. Understand how the
direction field for the first-order differential equation
y' = f(x,y) allows one to "see" the family of
solutions [Figures 1-4 page 513]. Be able to sketch the
direction field of y' = f(x,y) [Example 1, Exercises
9,10, Exercises 11-14 -choose a 5 by 5 grid of points centered on the
initial value]. Given a direction field be able to sketch
solutions, and be able to sketch the solution with a given
initial value [Figures 7,8 page 514; Example 2; Exercises
1a, 2a, 7, 8, 18].
Euler's Method. Given a first-order differential equation y'=f(x,y), an interval [0,T], an initial value y0 and a number N, be able to apply Euler's method with N steps to find approximate values y(T/N), y(2T/N), ... , y(T) for the solution satisfying y(0) = y0 [Example 3 page 517, Example 4 page 518, Exercises 21-24].
7.3 Separable equations. Understand what a separable equation
is and be able to solve it by integration, applying the
"separate, integrate, solve" method [Example 1a page 522,
Example 2 page 523]. Understand how the "+C" from
one of your integrals turns into the undetermined constant
in the general solution, and how an initial value determines
what that constant must be [Example 1b, Exercises 9-14].
Mixing problems. Be able to apply the "rate in - rate out" method to convert a mixing problem to a separable differential equation, and then be able to solve the equation [Example 5, problems 34-36].
7.4 Exponential growth and decay. y' = ky is an important
special case of a separable equation. The general solution
is y=Cekt, called "exponential growth" if
k is positive, "exponential decay" if k is
negative. Understand that C = y(0) [Exercises 1,2].
Be able to calculate k from C and one other
data point [Exercise 3].
Be able to calculate C and k from two data points
[Exercise 4]. Special case: the half-life. Be able to calculate
k from the half-life, and vice-versa [Example 3, Exercises
Compound interest. Understand that $1 invested for one year at annual interest t (in dollars) gives $1+t if compounded annually, $(1 + t/2)2 if compounded twice a year $(1 + t/n)n if compounded n times a year, and $et if compounded continuously [Example 4 page 536, Exercises 16a, 17a, 18].
Material from Appendix I (complex numbers) and Notes (second-order equations) will NOT be on this test.