Review Chapter 5 using Midterm 1 Review and Midterm 2 Review.

5.10 Understand the definition of an improper integral as a limit
of definite integrals (*Definitions on pp.424 and 427*). Understand that
the integral *converges* if the limit exists, and *diverges*
otherwise. *Examples 1,2,3* are
important. Understand the behavior of the `1/x ^{p}`
integrals from 1 to infinity: converge if

The Comparison Theorem will not be covered in this examination.

6.1 Be able to do a problem like *Example 2* (find area enclosed by
two curves): locate intersection points, set up integral with
correct sign (need to know which curve is higher), and evaluate
(*Exercises 7,9,13,15*). Be able to integrate with respect to
`y` when appropriate (*Example 5, Exercises 11,12*).
Be able to use numerical techniques (*Example 4, Exercises 21-24*).
Be able to handle curves that intersect twice or more (*Exercise 27*).
Be able to calculate areas enclosed by parametric curves (*Example 6,
Exercises 31, 35*).

6.2 Understand how to calculate a volume by the method of slices:
determine your axis of integration, find volume of infinitesimal
slice (`dV = A(x)dx` if `x` is the variable of
integration, `A(x)` the cross-section for
given `x`) and integrate `dV` between appropriate endpoints.
See "Definition of Volume," p.449. For solids of revolution the
axis of integration is the axis of symmetry (*Examples 1,2,3,4,
Exercises 1-6*). Know how to handle the situation when that axis
is parellel to but not equal to a coordinate axis (*Examples 5,6,
Exercises 13,14*). Understand how to set up integrals for more
general volumes (*Examples 7,8, Exercises 30,33,34,35*). Be
able to use the method of circular shells, especially when the
slice method leads to difficult equations or integrals (*Example 9,
Exercises 49,50,51b*).

6.3 Be able to set up the integral giving the length of the
graph of a paramtric curve `x = f(t), y = g(t)` between `t=a` and `t=b`.
Box 1 p.463, *Example 1, Exercises 4,5,6*. Understand the special
case where the curve is the graph of `y = f(x)` (you use `x`
as your parameter) *Examples 2,3,4, Exercises 3,7*. Since these
integrals are often difficult or impossible to solve by anti-differentiation,
be able to use numerical methods (*Example 2, Exercises 11, 12* (use
`n` = 3 or 4 if doing it by hand).

6.4 Understand the definition of the average value of a function on
an interval (Box p.468). (*Example 1, Exercises 5,11,12,13*).

6.5 Understand how the formula Work=Force x Distance becomes an
integral when the force varies over the distance. (*Examples 1,2,
Exercises 1,2,3,4*).
or when both force and distance vary during the problem (*Examples 3,4,
Exercises 10,11,17a,18*).

Be able to calculate the center of mass of a plate of constant
density (Figure, p.476) of the special type shown on p.478: it has the
shape of the area under the curve `y = f(x)` for `x`
running from `a` to `b`. See Box 12, p.479; A is the
total area of the plate. (*Example 7* - note use of symmetry to
simplify calculation; *Exercises 37-40*).

6.7 Understand that a probability density function `f(x)`
is a positive
function with total integral = 1, and how the integral of `f`
from `a` to `b` represents a certain probability
(that the value of the associated random variable will lie between
`a` and `b`). (*Example 1. Exercises 1,2,3*).
Understand that an exponentially decreasing probability density
function ` f(x) = 0` for `x` negative,
`f(x) = c e ^{-cx}` for

The normal distribution will not be covered in this examination.

Use the Chapter Review p.493 for further reviewing.

Concept check: 1-6,8,11,12.

Exercises 1-3 (area)

Exercises 4,5,6,7,10-14a (volumes)

Exercises 15-18 (arc length)

Exercises 19,20,21a (work)

Exercise 24 (center of mass = centroid)

Exercise 26 (average value of function)

Exercises 29, 31 (probability density functions)