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/xp
integrals from 1 to infinity: converge if p>1, diverge otherwise
(Box 2, p.427), Example 4.
Exercises 5-10 (be ready to use substitution!).
Also Exercises 13,14,18,19 which use
a preliminary integration by parts. Understand the behavior of the
1/xp integrals from 0 to 1: converge if p<1,
diverge otherwise. Examples 5,6,7.
Understand Example 8 - requires integration by parts: it means
that ln(x) gives a convergent integral
from 0 to 1.
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,
or when both force and distance vary during the problem (Examples 3,4,
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 x positive must have
the same coefficient c in both places (Example 2).
Understand the mean of a probability density function as
the expected value of the associated random variable (discussion, p.489)
and be able to calculate it (Example 3: the mean of the
exponentially decreasing p.d.f. given above is 1/c;
Example 4: work backwards from the mean, Exercises 4,5,6).
Understand the definition of median (p.490) and be able
to calculate it (Exercises 7,8).
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)