15 MULTIPLE INTEGRATION
15.1 Integration in Two Variables (LT Section 16.1)
Preliminary Questions
1. If S8,4 is a Riemann sum for a double integral over R = [1, 5] [2, 10] using a regular partition, what is the area of
each subrectangle? How many subrecta
MATH 31B: PRACTICE MIDTERM 2
JOE HUGHES
For each of the following five questions, indicate whether the statement is true or false.
1. Suppose that an = f (n) for some function f (x). If limn an exists, then limx f (x)
must also exist.
P
2. If
n=1 an conv
MIDTERM 2
Math 32B, Lecture 1 (11 am)
Nanak
University ID:
Discussion Section:
Tuesday Thursday
Casey Tianyu Mohammed Zuhair
Directions: The exam has ve questions and is worth 50 points total. No outside materials (books, notes,
phones, etc.) or calcula
Theorem Quiz Info
Math 32B, Week 10
There will be a theorem quiz in your discussion section in week 10. The theorem quiz will
cover the three major theorems of multivariable calculus that we have discussed so far: the
Fundamental Theorem of Line Integrals
Sample Questions for Final Exam
Math 32B, Fall 2012
These problems are intended to show you the sorts of problems that might appear on the
exam and the approximate level of diculty you can expect. Some parts of the actual exam
will probably be easier than
Info and Study Suggestions for Final Exam
Math 32B, Fall 2012
The nal exam will be cumulative. It will cover Chapters 15 and 16 of the textbook, except
Section 15.5. There will be somewhat more emphasis on material that did not appear on the midterm
exams
Sample Questions for Final Exam
Math 32B, Fall 2012
These problems are intended to show you the sorts of problems that might appear on the
exam and the approximate level of diculty you can expect. Some parts of the actual exam
will probably be easier than
MIDTERM 1
Math 32B, Lecture 2 (2 pm)
"i » H .
Name: . jmigéMfflngX-x-
University ID:
TA Name:
Directions: The exam has ve questions and is worth 50 points total. N0 outside materials (books, .
notes, etc.) or calculators are allowed. Write your answers
1. Use a double integral to nd the volume under the graph of at, y) m 4 over the region-bounded by
the parabolas y = im? and y x 5 m 292. Include a sketch of the region of_integration in the nay-plane.
(10 paints)
nkaew {ame ak . . . . . . . . .
iw K
MATH 31B: MIDTERM 1 REVIEW
JOE HUGHES
1. Inverses
1. Let f (x) =
1
x3 .
Find the inverse g(x) for f .
Solution: Setting y = (x 3)1 and solving for x gives
yx 3y = 1
and
x=
Therefore g(x) =
1 + 3y
y
1+3x
x .
2. Let f (x) = x4 + 32x. Find a domain on which
MATH 31B: PRACTICE MIDTERM 1 SOLUTIONS
JOE HUGHES
For each of the following five questions, indicate whether the statement is true or false.
1. sin1 (sin(x) = x for every real number x.
Solution: This is false, because the range of sin1 (x) is [ 2 , 2 ].
MATH 31B: MIDTERM 2 REVIEW
JOE HUGHES
1. Trigonometric Integrals
1. Evaluate
R
sin4 x dx.
Solution: The key idea for evaluating trig integrals with even powers of sine and cosine is
to use the double angle formulas for cosine. In this case, the relevant i
MATH 31B: PRACTICE MIDTERM 1
JOE HUGHES
For each of the following five questions, indicate whether the statement is true or false.
1. sin1 (sin(x) = x for every real number x.
2. If f (x) is invertible, then so is f (x)3 .
3. If limx f (x) = and limx g(x)
MATH 31B: PRACTICE MIDTERM 2 SOLUTIONS
JOE HUGHES
For each of the following five questions, indicate whether the statement is true or false.
1. Suppose that an = f (n) for some function f (x). If limn an exists, then limx f (x)
must also exist.
Solution: