Math 3B
Checklist for Exam 1
You are responsible for knowing the material from the lectures and homework exercises.
Below is a checklist to highlight the main ideas that you may run into on Exam 1. You are
responsible for reviewing the lecture notes and knowing what is the acceptable level of work to
be shown on the exam. Be sure to use correct mathematical notation. Correct answers
without proper mathematical justification are not acceptable.
Sect 5.5 U-Substitution
1) Be able to evaluate indefinite integrals using the method of u-substitution. Remember
that indefinite integrals produce a family of antiderivatives that are vertical shifts of one
another. So, don’t forgot your constant of integration.
2) Be able to evaluate definite integrals using the method of u-substitution.
3) For a definite integral, give the exact value, not a decimal approximation. I recommend
that if you change variables from x to u, then switch the limits of integration to u values
also and finish the problem with variable u.
4) The method of substitution is the first method to consider when evaluating definite or
indefinite integrals.
Sect 6.1 Areas Between Curves
1) Be able to find the area between two curves by determining which function plays the
role of 𝑦!"# and 𝑦%"! .
2) Be able to draw the area of the region of interest and be sure to shade in the area on
your paper.
3) Some problems will give a specific interval and some problems do not. If no interval is
specified, then the area of interest, is the area that is bounded by ALL given curves that
create a closed and finite area.
4) Be able to find area using integration with respect to x or y. Choose the variable of
integration that will hopefully reduce your calculations.
5) Be prepared to graph basic functions from algebra, trig and precalculus classes.
Sect 6.2 Volumes
&
1) Know how to find volumes using the definition 𝑉 = ∫' 𝐴(𝑥)𝑑𝑥 . Be able to draw the solid.
2) Know how to find the volume of a solid of revolution whose cross-sectional slices are disk,
using the Disk method. The area function A(x) will be based on the area of a circle, 𝜋𝑟 ( . Be
able to draw the region and the solid. Be able to perform integrations with respect to both
variable x and y. Be able to do problems where the axis of rotation is not the coordinate
axes. If the cross-sectional area is a disk and the axis of rotation is a horizontal line, then we
will integrate with respect to x. If the cross-sectional area is a disk and the axis of rotation is
a vertical line, then we will integrate with respect to y. Remember that the radius is always
measured from the axis of rotation, AOR!
3) Know how to find the volume of a solid of revolution whose cross-sectional slices are
washers, using the Washer method. The area function A(x) will be based on the area of a
(
(
larger circle minus the area of a smaller circle, 𝜋𝑟")!*+
− 𝜋𝑟,--*+
. Be able to draw the
region and the solid. Be able to perform integrations with respect to both variable x and y.
Be able to do problems where the axis of rotation is not the coordinate axes. If the crosssectional area is a washer and the axis of rotation is a horizontal line, then we will integrate
with respect to x. If the cross-sectional area is a washer and the axis of rotation is a vertical
line, then we will integrate with respect to y. Remember that the radii are always measured
from the axis of rotation, AOR!
4) Know how to find the volume of a solid over a circular base or triangular base as in HW
problems.
Sect 6.3 Volumes by Cylindrical Shells
1) Be able to find volumes of solids using the cylindrical shells method. Be able to draw
the region and the solid. Be able to perform integrations with respect to both variable x
and y.
2) Be able to do problems where the axis of rotation is not the coordinate axes.
3) When using the method of cylindrical shells, if the axis of rotation is a horizontal line,
then we will integrate with respect to y.
4) When using the method of cylindrical shells, if the axis of rotation is a vertical line, then
we will integrate with respect to x.
5) Remember to integrate on only one side of the axis of rotation.
6) Average radii are measured from the axis of rotation.
Sect 6.4 Work
1) Know the definition of Work for a constant force used to displace an object. ( W = Fd )
2) Know the definition of Work for a non-constant force used to move an object from a to
&
b is 𝑊 = ∫' 𝐹(𝑥)𝑑𝑥 .
3) Be familiar with Hooke’s Law for finding forces needed to stretch a spring. Be able to
find the spring constant k, if needed. ( F(x) = kx ).
4) Know how to compute the work to move water out of a tank. In such problems, I will
𝑓𝑡
give you the acceleration due to gravity, if needed. 𝑎 = 9.8 𝑚5𝑠 ( or 𝑎 = 32 5𝑠 ( .
5) I will also provide you with density of water or any other liquid in the tank. The
𝑘𝑔
Density of water is 1000 5𝑚. or I will provide you with the Force density. For
example, the force density for water is 𝜌 = 62.4 𝑙𝑏5𝑓𝑡 . . The force density already
contains the acceleration due to gravity!!!
6) It is required that you label your drawing with a vertical axis and be sure to label clearly
where your axis starts with zero. Be sure to show how you come up with the work
expression for one chunk of water. Be sure to label your answer with units of work. Be
careful with your labels.
Sect 6.5. The Average Value of a Function and the Mean Value Thm
0
&
1) Know the definition of the average value of a function, 𝑓'/* = &1' ∫' 𝑓(𝑥)𝑑𝑥 .
2) Know the Mean Value Theorem for Integrals. Be sure you can draw the rectangle that
satisfies the conclusion of the Mean Value Theorem for Integrals. The height of the
rectangle is given by 𝑓'/* and the width is the length of the interval, [ a, b].
3) Be able to find the value, call it c, that makes 𝑓'/* = 𝑓(𝑐).
Sect 7.1 Integration by Parts
1) Know the method of integration by parts. It is used for integrating products, lnx and
𝑡𝑎𝑛10 𝑥, 𝑠𝑖𝑛10 𝑥 𝑎𝑛𝑑 𝑐𝑜𝑠 10 𝑥, . To integrate lnx and these three inverse trig functions,
you must know their derivatives!!!!
2) Use LIATE to help you determine which function plays the role of f(x) or u.
3) Be able use integration by parts to evaluate definite integrals too.
4) Remember to consider u-substitution method first before using integration by parts.
Sect 7.2 Integrating Trig Functions (This material is pushed to Exam 2)
1) Be able to integrate ∫ 𝑠𝑖𝑛- 𝑥𝑑𝑥 , where n is a natural number. For 1 ≤ 𝑛 ≤ 3, it is
reasonable to integrate without the Reduction formula. Know how to integrate 𝑠𝑖𝑛( 𝑥
using the double angle formula. Be sure to know your double angle formula.
𝑠𝑖𝑛! 𝑥 =
"#$%& (!))
!
2) Be able to integrate ∫ 𝑐𝑜𝑠 - 𝑥𝑑𝑥 , where n is a natural number. For 1 ≤ 𝑛 ≤ 3, it is
reasonable to integrate without the Reduction formula. Know how to integrate 𝑐𝑜𝑠 ( 𝑥
using the double angle formula. Be sure to know your double angle formula
1 + cos (2𝑥)
𝑐𝑜𝑠 ( 𝑥 =
2
3) Be able to integrate products of powers of sine and cosine. ∫ 𝑠𝑖𝑛- 𝑥𝑐𝑜𝑠 2 𝑥𝑑𝑥 . We
discussed three methods that cover all 4 cases. To do such problems, requires using the
Pythagorean Identity, 𝑠𝑖𝑛( 𝑥 + 𝑐𝑜𝑠 ( 𝑥 = 1
4) Be able integrate ∫ 𝑡𝑎𝑛- 𝑥𝑑𝑥 , where n is a natural number. For 1 ≤ 𝑛 ≤ 2, it is
reasonable to integrate without the Reduction formula.
5) Be able integrate ∫ 𝑠𝑒𝑐 - 𝑥𝑑𝑥 , where n is a natural number. For 1 ≤ 𝑛 ≤ 2, it is
reasonable to integrate without the Reduction formula.
6) Be able to integrate products of powers of sine and cosine. ∫ 𝑡𝑎𝑛- 𝑥𝑠𝑒𝑐 2 𝑥𝑑𝑥 . To do
such problems, requires using the Pythagorean Identity, 𝑡𝑎𝑛( 𝑥 + 1 = 𝑠𝑒𝑐 ( 𝑥.