Fall 2008, Math 152, sections 201-202, Honors
Exam 1 Overview
Exam I is on Thursday, February 14, 7:30-9:30 pm., in RICH 114. It covers sections
6.1-8.4. It is assumed that you know the material of Chapters 1-5 from Calculus I.
To prepare for Exam 1: study your lecture notes, read the textbook (at the very least each
chapter overview), study all quizzes (not to repeat same mistakes), and solve as many recommended
problems as you possibly can. Past common exams and Challenger 1 give you more problems to
practice on. Good Luck! Gig’em Aggies!
1. Chapter 6. Integration. (sections 6.1-6.6). You need to know and be able to apply the
following:
a) Definitions of: an antiderivative, an indefinite integral, Riemann sum, definite integral.
Geometric and mechanical meaning of integrals.
b) Table of indefinite integrals, page 458.
c) Properties of indefinite and definite integrals.
d) Fundamental Theorem of Calculus
e) Substitution rule for indefinite and for definite integrals.
Suggested problems:
Page 416: 1-21, 24-58 (even), 61, 66, 68, 70, 72, 76, 77, 78, 80, 82, 86, 91, 92
2. Chapter 7 (7.1-7.5). Applications.
a) Area between curves- integral of (variable) height. Be able to choose the best method of
integration (integrating with respect to x or y?). Be able to split the region into two or more
parts, if needed.
b) Volume of a solid is an integral of the cross-sectional area (area function may be given or
you should be able to compute using the information given.)
c) Volume of a solid of revolution:
disc method (cross sections are discs perpendicular to the axis of revolution) - the volume
by this method is the integral of area πR2 of a cross-section.
washer method (cross sections are discs with holes, perpendicular to the axis of revolution)
- the volume by this method is
the integral of the difference of areas, πR2 − πr2 , for a cross-section.
shell method - the volume by this method is the integral of surface area of a cylindrical
shell generated by rotating a thin rectangle about the axis of revolution.
Z
V =
b
2π(radius)(height)(thickness)
a
where radius is the distance from a thin rectangle generating a shell to the axis of revolution,
height is the height of this rectangle and thickness of the rectangle is either dx (if rotation
is about y−axis or about a line parallel to it) or dy (if rotation is about x−axis or about a
line parallel to it).
d) work done by a variable force: If F = F (x) acts along a straight line on a particle moving
Rb
along the line from x = a to x = b , then W = a F (x) dx. In particular, be able to solve the
following types of problems:
spring problems Here F = kx whereRx is the displacement measured from natural length, k
b
is the spring constant. Thus W ork = a kx dx, so if you know force or work for some stretch,
you should be able to find the spring constant and then you can answer any question about
the spring.
problems on
R b pumping
R bwater out of tanks or on lifting variable weights:
work=W = a Wslice = a (weightslice )(distanceslice to top)
Rb
1
e) average value of a function on a closed interval: fave = b−a
a f (x)dx
f) Mean Value Theorem for integrals
Suggested problems: Page 452: 1, 4, 6, 7, 10, 12, 15,16, 18,19, 21, 23, 26, 28, 29, 30, 31,
32, 34. If you wish to challenge yourself further, try some Problems Plus at the end of the
Chapter.
3. Chapter 8 (8.1-8.4) You need to know the following integration techniques:
a) integration by parts:
Rb
R
R
Rb
udv = uv − vdu, a udv = uv|ba − a vdu
trigonometric
functions:
R
R
Rb) integrating
sinm x cosn x dx, tanm x secn x dx, sin mx cos nx dx, etc.
relationships used:
sin2 x + cos2 x = 1, sin 2x = 2 sin x cos x, cos 2x = cos2 x − sin2 x, sin2 x =
2x
cos2 x = 1+cos
, tan2 x + 1 = sec2 x, product to sum formulas, etc.
2
1−cos 2x
,
2
c) integrating
using trigonometric
substitutions:
√
√
√
for a2 − x2 use x = a sin x; for a2 + x2 use x = a tan x; for x2 − a2 use x = a sec x.
d) integrating rational functions using the method of partial fraction decomposition..
Step 1. If a rational function is improper, use long division to transform the function into a
polynomial and a proper fraction.
Step 2. Decompose the proper fraction into partial fractions with undetermined coefficients.
Step 3. Find the undetermined coefficients. Step 4. Integrate the sum of the partial fractions.
Suggested problems:
Page 518: 1-4, 9-42