## MAT126 Review for Final

Review Chapter 5 using Midterm 1 Review
and Midterm Review.
5.9 Understand the definition of an improper integral as a limit
of definite integrals (*Text on pp.427-428*). Understand that
the integral *converges* if the limit exists, and *diverges*
otherwise (*Definition 1 on p.428*). *Examples 1,2,3* are
important. Understand the behavior of the `1/x`^{p}
integrals from 1 to infinity: converge if p>1, diverge otherwise.
*Exercises 5,6,7,8* (be ready to use substitution!).
Also *Exercises 11,14,17,18* which use
a preliminary integration by parts. Understand the behavior of the
`1/x`^{p} integrals from 0 to 1: converge if p<1,
diverge otherwise (*Definition 3 on p.431*). *Examples 5,6,7*.
Understand Example 8 - requires integration by parts. *Exercises
23,24,25,26*. Understand that `ln(x)` gives a convergent integral
from 0 to 1. Comparison Theorem and applications 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 5,11,13,15*. 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 25*).
Areas enclosed by parametric curves will not be covered
in this examination.

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, and integrate `dV` between appropriate endpoints.
See "Definition of Volume," p.457. For solids of revolution the
axis of integration is the axis of symmetry (*Examples 2,3,4,
Exercises1-6*). Understand how to set up integrals for more
general volumes (*Examples 5,6, Exercises 20,21,24,26,28*). Be
able to use the method of circular shells, especially when the
slice method leads to difficult equations or integrals (*Example 7,
Exercises 41,42*).

6.3 Be able to set up the integral giving the length of the
graph of `f(x)` between `x=a` and `x=b`.
*Examples 2,3,4, Exercises 5,6,7*.

6.4 Understand the definition of the average value of a function on
an interval (Box p.470). *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 *Example 3,
Exercises 7,11,13a,14*

to come

Use the Chapter Review for further reviewing.

Concept check: 1-7.

Exercises 1-4 (area)

Exercises 5,8,9-12a (volumes)

Exercises 13,14 (arc length)

Exercises 16, 17a (work)