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/x ^{2}` on page 428 and

B: Function goes to infinity at a finite value

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
total work
`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 `y _{0}` and a number

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=Ce ^{kt}`, called "exponential growth" if

Compound interest. Understand that $1 invested for one year at annual interest t (in dollars) gives $1+t if compounded annually, $(1 + t/2)

Material from Appendix I (complex numbers) and Notes (second-order equations) will NOT be on this test.