MATH 52 FINAL EXAM SOLUTIONS (AUTUMN 2003)
1. Evaluate the integral by reversing the order of integration
1 0 3 3y
ex dxdy. Solution.
1 0 3 3y
2
e dxdy =
0 3 0
x2
3
x/3
ex dxdy ex y
0 3
2
2
= 1 2 = ex 6
y=x/3 y=0 9
3
dx =
0
x x2 e dx 3
Assignments for MTH 52
WEEK #3 (Due: Oct 15th)
Problem 1.
Here were using the so-called Parallel Axis Theorem, which tells us that the moment of
inertia I(x,y) about any point (x, y) is equal to I(,) + mr2 , where I(,) is the moment of
xy
xy
inertia about
Math 40, Section
Homework 4
Feb 2, 2015
Section 2.2 - 18, 20, 23*, 30, 42, 47* , Section 2.3 - 8, 16
Worked with:
2.2.18 Show that matrices A and B are row equivalent, and nd a sequence of elementary row
operations that will convert A into B.
2 0 1
3 1 1
Assignments for MTH 52
WEEK #8 (Due: November 19th )
5
START
4
Problem 1. In the count before you jump downhill slalom you
need to go from the start to nish around the red ags (see the
picture on the left). The direction of going around the ags is not
spe
Assignments for MTH 52
WEEK #7 (Due: November 12th )
(
) (
)
2
2
Problem 1. Let F = 2xy ex + 1 i + ex + x j .
Find
F T ds where C is the upper part of the circle x2 + y 2 = 4 oriented from left to
C
right.
Hint: Add the diameter contained in the xaxis a
Assignments for MTH 52
WEEK #9 (Not graded )
Problem 1. Compute the work of the force F = (x + z) i + (x y + 2z) j + (y 4x) k
acting along the perimeter of the triangle ABC with vertices A (1, 0, 0), B (0, 1, 0) and
C (0, 0, 1). Assume that the path start
Assignments for MTH 52
WEEK #1 (Due: Oct 1st )
Material covered: 13.1 Double Integrals.
13.2 Double Integrals over More General Regions.
Changing the order of integration.
Problem 1. Give an estimate on the volume of the sand in 3 5m rectangular sandbox i
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Assignment for Math 52
WEEK #1
Material covered:
5.1 Areas and Volumes.
5.2 Double integrals: over rectangle, over more general regions, Fubinis theorem.
Section 5.1
7. a. The iterated integral from slicing the region with parallel planes of the from
x =
Assignments for MATH 52
WEEK #4
Section 5.8
20(a). The shell is a spherical coordinate box, with 3 d = 4, 0 , 0 2.
2
Mass =
4
(.122 )(2 sin()ddd
dV =
0
0
3
Integrating rst with respect to and we get
4
4 d = .48
(.24)(2)
3
45 35
= 74.976 grams.
5
1
(b) The
5.5.8(a)
y
+2
2
1
y
2
0
y
+2
2
1
2
(x xy)
(2x y) dxdy =
1
4 dy = 4.
dy =
y
2
0
0
(b) The region of integration D can be seen to be a parallelogram with (x, y) vertices
(0, 0), (2, 0), (1/2, 1), (5/2, 1). The transformation u = 2x y, v = y corresponds D wi
HW 8 Solutions
May 26, 2011
Exercise 7.1 Problem 4: Let X(s, t) = (s2 cos t, s2 sin t, s), |s|
0 t 2.
a) Find a normal vector at (s, t) = (1, 0)
b) Determine the tangent plane at the point (1, 0, 1).
c) Find an equation for the image of X in the form F (x
MATH52: SPRING 2011
HOMEWORK 7 SOLUTIONS
1. 6.2, 23
Let D be a region to which Greens theorem applies and suppose that u(x, y) and
v(x, y) are two functions of class C 2 whose domains include. Show that
D
(u, v)
dA =
(x, y)
(u v) ds,
C
where C = D is orie
MATH52 : SPRING 2009
HOMEWORK 2
5.3, 8
Sketch the region of integration, reverse the order of integration, and evaluate both
iterated integrals for the following:
2
0
cos x
sin x dy dx.
0
Solution. The region of integration for this problem is given in Fi
MATH 52, SPRING 2011
HOMEWORK 9 SOLUTIONS
Section 7.3 : # 6, 12, 14, 20, 28
(6) We have div(F ) = 3. So the triple integral side of Gauss theorem is
cylindrical coordinates this becomes
9r 2
2
3
2
3(9 r2 ) rdrd =
3dz rdrd =
0
0
0
0
D
3 dV . In
243
2
The b
HW 5 Solutions
April 11, 2011
Exercises 1.4 Problem 16: Compute the area of the triangle having vertices A = (1, 1), B = (1, 2), and C = (2, 1).
Solution: The vector BA is
2 1
2
1 1
=
. The vector CA is
=
1 1
1
21
3
. We may think of these vectors in R2 a
Assignments for MTH 52
WEEK #5 (Due: October 29th )
Problem 1. Find the x and y coordinates of the centroid of the surface cut from z =
x2 + y 2 by the surface (x a)2 + (y a)2 = a2 .
(the integral giving the z coordinate is too hard to compute.)
Problem 2
Assignments for MTH 52
WEEK #2 (Due: Oct 8th )
Material covered: 13.3 Area and Volume by Double Integration.
Determinant and area/volume.
13.4 Double Integrals in Polar Coordinates.
Improper integrals.
13.5 Applications of Integration (two dimensional).
P
Assignments for MTH 52
WEEK #3 (Due: Oct 15th )
Problem 1. Let uniform rectangular plate with dimensions a b and mass m = ab be
centered at the origin.
In the last homework you have computed that
I0 =
)
1 ( 2
m a + b2
12
Use that result and the theorem ab
MATH 52 FINAL EXAM
1. (a) Sketch the region R of integration in the following double integral.
1 0 1
xey dy dx
x
5
(b) Express the region R as an x-simple region. (c) Evaluate the integral by changing the order of integration. 2. (a) Let T be a s
MATH 52 MIDTERM 1 SOLUTIONS (AUTUMN 2003)
1. Evaluate the following double integrals. (a) Evaluate the double integral
1 0 x 4-x
2y dy dx Solution.
1 0 x 4-x 1 4-x 1 1
2y dy dx =
0
y
2 x
dx =
0
16 - 8x dx = 16x - 4x
2 0
= 12
(b) Sketch the r
MATH 52 SAMPLE MIDTERM
October 16, 2004 Name: Numeric Student ID: Instructor's Name: I agree to abide by the terms of the honor code: Signature: Instructions: Print your name, student ID number and instructor's name in the space provided. During the
MATH 52 SAMPLE MIDTERM II
November 13, 2004 Name: Numeric Student ID: Instructor's Name: I agree to abide by the terms of the honor code: Signature: Instructions: Print your name, student ID number and instructor's name in the space provided. During
Math 52 - Fall 2007 - Midterm I
Name:
Student ID:
Signature:
Instructions:
Please print your name and student ID. Your signature indicates that you accept the honor code.
During the test, you may not use notes, books, calculators or telephones. Read each
Math 52- Winter 2010 - Midterm Exam I
Please circle the name of your TA:
Jack Hall
Xiannan Li
Circle the time your TTh section meets: 10:00
David Sher
11:00
1:15
2:15
Your name (print):
Student ID:
Please sign the following:
On my honor, I have neither gi
Math 52 - Autumn 2006 - Midterm Exam I
Problem 1.
2
2
cos(x2 ) dx dy
a) (10 points) Sketch the region of integration of
0
(0,2)
y
y=x
(2,0)
2
2
cos(x2 ) dx dy
b) (10 points) Evaluate
0
y
Solution: After the change of order of integration we get:
2
x
2
x c
MATH 52 MIDTERM I OCTOBER 14, 2009
THIS IS A CLOSED BOOK, CLOSED NOTES EXAM. NO CALCULATORS OR OTHER
ELECTRONIC DEVICES ARE PERMITTED.
THERE IS ONLY ONE INTEGRAL EVALUATION, IN PROBLEM 3(a). THERE ARE 5
PROBLEMS WORTH 10 POINTS EACH, ON 7 PAGES (INCLUDING
Math 52 - Autumn 2010 - Midterm Exam I
Name:
Student ID:
Signature:
Instructions:
Print your name and student ID number, select your section number and TAs name,
and write your signature to indicate that you accept the Honor Code.
There are 5 problems o
MATH 52 MIDTERM I APRIL 22, 2009
THIS IS A CLOSED BOOK, CLOSED NOTES EXAM. NO CALCULATORS OR OTHER
ELECTRONIC DEVICES ARE PERMITTED.
YOU DO NOT NEED TO EVALUATE ANY INTEGRALS IN ANY PROBLEM. THERE
ARE 4 PROBLEMS, EACH WORTH 10 POINTS.
Please sign the foll
Math 52 - Winter 2006 - Midterm Exam I
Problem 1. (8 pts.) Let R be a region of R2 symmetrical about the y -axis, and R+ be
the half of the region R contained in the half plane x 0. Mark as TRUE/FALSE the
following statements:
dA = 2
a)
dA
TRUE
R+
R
x dA