Math 242C
Midterm Exam 3 Review Sheet
Practice exams
Here are some exams I have given in the past years covering roughly
the same material. There are some differences in both the syllabus and in
the notation and terminology, since some of the earlier exams
were based on a different text (by Stewart) or a different edition of the
Edwards/Penney text we are using now. You can ignore (for now)
problems asking for a "surface area", since this material, which
corresponds to Section 13.8 in Edwards/Penney, is not on the exam
syllabus.
In terms of the difficulty level and nature of the problems, these
exams are pretty representative of what you can expect in our exam.
General Information
- Review session: I will hold an exam
review session Monday, November 27, 5 pm - 6 pm, Room 245
Altgeld.
- Exam date and time:
The exam will be given during the regular class time,
Wednesday, 11/29/2006, 4 - 5 pm.
- Exam location: The location will be the same
as for the previous midterms, 112 Greg Hall.
Greg Hall is located on the east side of Wright Street, near the
library.
Note: Seats will again be assigned, in the same way as for
the last midterm. In particular, if the exam room is unoccupied
during the 3 pm - 4 pm slot, we'll start passing out seat assignments
at around 3:45.
- Exam rules: Same as usual. Academic
honesty is expected, and cheating will not be tolerated. Follow the
Aggie Code of
Honor. No earphones, ipods, blackberries, laptops, walkmans, and
similar devices. Student IDs may be checked.
- Calculator policy: As with all exams and quizzes in this class,
calculators are not allowed.
The problems will be such that a calculator
is not needed and would not be of any help.
- Books/notes policy: This is a closed books/notes exam.
Books, classnotes, formula sheets, etc., are not allowed, and must be
securely stowed away without easy reach (e.g., inside a closed backpack).
Exam content:
The exam will cover Sections 11.8 (cylindrical and spherical
coordinates),
and Chapter 13 through 13.7 (double and triple integrals) except
for those parts that have not been covered in class and explicitly marked
in the lecture summaries
as material that can be skipped. (Sections 13.8 and 13.9 will
not be on this exam.)
Most students find Chapter 13 (double and triple integrals) to be the
most difficult of the Math 242 syllabus. The only way to get
comfortable with those types of problems is through lots of practice,
something that cannot be emphasized enough. In contrast to
some of the earlier material which depended more on knowing the right
formulas, just memorizing some formulas is not going to cut it here.
You have plenty of material to practice with: Some two dozen examples
worked out in detail in the lectures and the discussion sections,
some forty homework problems covering Sections 13.2 - 13.8.
In working the problems on your own, follow the methods from class and
the discussion sections. In particular,
For double and triple integrals the set-up is the key, and represents
80 percent of the intellectual work. Once you have a multiple integral
correctly set up as an iterated integral with appropriate integration
limits, the evaluation is a routine, though sometimes lengthy and
timeconsuming, calculation at the level of a Calc 2 exercise. Most of
the exam problems on multiple integrals will likely just ask for the
set-up of an appropriate double or triple integral and can skip the
evaluation step. (See the practice exams for examples.)
Grading
Partial Credit:
"Quickie" type questions, and true/false or multiple choice questions,
are all-or-nothing
problems; you either know or don't know the relevant formula, and
giving partial credit for these problems would not make much sense.
For other problems, partial credit will be given if you
make significant progress towards the solution.
The problems will be largely independent of each other, so if you can't do one
problem, it will have no effect on the other problems.
Curve: I do not use a fixed grading scale (such as 90 points = A,
80 points = B, etc.), but rather set a curve after each exam and quiz,
depending
on the performance of the class. The curve used will be announced on the
course web page, and your computer grade reports will show both your raw
and curved scores.
Grading: The exam will be graded by the end of the week
and returned in the discussion sections on Tuesday following the exam.
Scores should be online by Saturday evening. Check the course webpage
periodically; I will make an announcement there when the scores are
online, and provide a link to access the online scores.
As a reminder, each of the three hour exams
counts 15% towards your course grade; the quizzes count 25%
(one drop score);
and the Final Exam counts 30%. See the Course Information Sheet for details on
the grading policy.
Missed Exam policy. If you miss the exam, but have a valid
excuse (e.g., illness), I will count the exam as excused. The absence
must be documented by a letter from the Dean's Office (300 Student Services
Building, 610 East John St.). If you know ahead of time of the
absence, let me know beforehand (email ajh@uiuc.edu), and see someone
in the Dean's office to explain the situation and arrange for a Dean's
letter. If the absence is due to unforeseen circumstances (such as
illness), get in touch with me and with the Dean as soon as possible
after the exam. A missed exam without a valid excuse counts as 0
points.
Detailed syllabus
Section 11.8 Cylindrical and spherical coordinates
- Definition, geometric interpretation
- Conversion formulas between cylindrical, spherical, and
rectangular coordinates
- Conversion of equations in rectangular coordinates to cylindrical
and spherical coordinates
- Surfaces and regions in cylindrical and spherical coordinates:
cylinders, spheres, cones, planes, half-planes
Note on trig values: Converting between coordinate systems
often requires computing sines and cosines. Calculators are not
allowed in quizzes and exams, but the problems will be such that you
do not need a calculator to compute the values of sines and cosines.
You should know the values of trig functions at 0, Pi/6, Pi/4, Pi/3, Pi/2,
etc.
Skip:
Last part of 11.8 (p.844 through end), on longitudes and latitudes.
Examples: 1 - 7
Homework (not to be turned in):
1, 5, 7, 9, 17, 23, 27, 29, 31, 33, 35, 39, 43, 45, 47, 49, 51
Section 13.1:
Double integrals
- Computation of double integrals over rectangular regions
Skip: There won't be any quiz/exam problems on Riemann sums
(p. 942/943), or the relation between double integrals and
cross sections (p. 946 middle through end of section). However,
you should take a look at the pictures illustrating these concepts, as
they help motivate the concept of a double integral and its
application to the computation of volumes (which will come up in a
later section).
Examples: 2, 3, 4
Section 13.2:
Double integrals over general regions
- Computation of double integrals over general regions R
- Vertically simple region (constant limits on x, variable limits
on y)
- Horizontally simple region (constant limits on y, variable limits
on x)
- Reversing order of integration
Examples: 2, 3, 4
Homework:
1, 11, 13, 15, 19, 31, 33
Section 13.3:
Area and volume by double integrals
- Computation of volume via double integrals
- Computation of area via double integrals
Examples: 1 - 4
Homework:
3, 7, 11, 15, 19, 27, 29, 35
Section 13.4:
Double integrals in polar coordinates
- Area element dA in polar coordinates
- Computing double integrals in polar coordinates
- Converting double integrals from rectangular coordinates
to polar coordinates
- Application:
Evaluation of the "Gaussian integral", the integral of exp(-
x2) from -infinity to infinity
Examples: 1, 2, 3, 4, 5
Homework: 3, 9, 13, 17, 27, 29, 34
Section 13.5:
Applications of double integrals
- Mass, mass density, mass element dm
- Centroid (center of mass)
- Moments of inertia
Ix,
Iy,
I0 (polar moment of inertia)
Skip: First and second theorem of Pappus
(p. 973 - 975),
formulas for kinetic energy and radius of gyration ((9), (10), (11)).
Examples: 1, 2, 3, 9, 10
Homework: 7, 15, 27, 31, 33
Section 13.6:
Triple integrals
- Triple integrals in rectangular coordinates
- Volume by triple integrals
- Mass, mass density
- Center of mass (centroid)
- Moments of inertia Ix, Iy, Iz
Examples: 1, 2
Homework: 5, 9
Section 13.7:
Triple integrals in cylindrical and spherical coordinates
- Review (from 11.8):
Definition and geometric interpretation of cylindrical and spherical
coordinates, conversion formulas
- Volume element dV in cylindrical and spherical coordinates
Examples: 1, 2, 3, 4
Homework: 1, 7, 8, 9, 19, 21, 25, 33, 37, 39
Typical tasks
Below is a list of typical computational problems from Chapter 13 and
Section 11.8. Most exam problems will be of one of these standard
types.
- Conversion between rectangular, cylindrical, and spherical
coordinates (11.8)
- Iterated integrals (13.2)
- Evaluation of
iterated integrals by reversing order of integration (13.2)
- Areas by double integrals (13.3)
- Double integrals in polar coordinates (13.4)
- Volumes by double integrals (13.3)
- Volumes by triple integrals (13.6, 13.7)
- Mass, center of mass (centroid), and moments of inertia,
of a lamina (13.5)
- Mass, center of mass, and moments of inertia,
of a solid object (13.6, 13.7)
- Triple integrals in cylindrical and spherical coordinates (13.7)
Miscellaneous hints and comments
- Computing multiple integrals:
The most difficult part in many of these problems is
setting up a double or triple integral
as an iterated integral with appropriate limits on each of
the individual integrals. To this end, you need to express the domain
R of integration in terms of inequalities on the variables x, y
(and z in the case of 3-dimensional integrals). Don't
try to derive these inequalities in the abstract,
but sketch the domain R
and ``read off'' the appropriate inequalities from the picture.
For integrals over rotationally symmetric objects (cones, etc.),
a z-r plot (depicting a vertical cross section of the object)
is often helpful. Last, but not least,
remember the number one rule for setting up multiple integrals:
The limits in the outer integral must be constant, and those
in the middle integral can only involve the outer variable.
- Choosing appropriate coordinates:
Sometimes you don't have a choice. A problem may specifically ask you
to do an integral using cylindrical coordinates
or spherical coordinates
(or do it both ways!). At other times, a problem may include a hint as
to what coordinate system to use. However, if you are faced with integral
and have to decide on a coordinate system, here are some general
guidelines: First, the shape of the region of
integration often makes the choice obvious.
If the integral is over a spherical region,
try to use spherical coordinates. When the integral is over
a rectangular or triangular region, use rectangular coordinates.
For cone shaped regions, either spherical or cylindrical coordinates
should be used; which one depends on the particular case.
Second, look for some tell-tale signs in the function to be
integrated: a term x2 + y2 suggests using polar
(in 2 dimensions) or cylindrical (in 3 dimensions) coordinates;
a term x2+y2+z2 suggests using
spherical coordinates.
- Integration techniques:
You should know how to integrate by substitution and by parts.
(Hint: When doing an integration by substitution,
make sure that you also substitute the limits of
integration.) Some trig integrals (such as the integral of (cos
x)3 (sin x)
can be done
with a simple trig substitution (u = cos x in this case), and you should be
able to do such integrals.
However, you are not expected to be able to compute
more esoteric integrals such as the integral of (sin x)3,
or the integral of Squareroot[1-x2]; if you run into one of
these integrals, it is an indication that you
did something wrong and should try a different approach
(for example, changing to different coordinates, or reversing the order
of integration).
You should know how to handle integrals over
sin2x and
cos2x. (Hint: in each case there is a trig identity that
converts the expression into a simple sine or cosine (without the
square).)
- Trig functions: When evaluating an integral
in polar, cylindrical, or spherical coordinates, you may need to
kow some special values of sine or cosine. Aside from the values at
multiples of pi/2 (which are always 0, 1, or -1),
you should know the values of sine and cosine
at pi/6, pi/4, pi/3 (corresponding to angles of 30,
45 and 60 degrees): at these three angles the sine
increases from 1/2 to Squareroot[2]/2 (or
1/Squareroot[2]) to Squareroot[3]/2, while the cosine goes through the same
three values in reverse order. From the values at pi/6,
pi/4 and pi/3, you can derive values at angles 2 pi/3, 3 pi/4,
etc., using standard identities for trig functions.
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