MATH2004C Assignment 2
Last Name: First Name:
Student ID:
• You may either write your answers on a copy of this assignment, or on your own
paper or on your electronic devices (you do not need to copy the questions).
• Due date: Before 11:59 pm on Tuesday, March 8th.
• The assignment has 10 questions and is out of 40 points. Each question is
out of 4.
• No email submission will be accepted.
• No excuses regarding technical issues will be accepted. It is your responsibility
to double check that you submitted the right file. Don’t wait until too close to
the deadline to start working.
• Submission Requirements: Submit your work as only one .pdf document.
Files in format different then .pdf or that are not in one document will not
e marked and you will obtain 0. Submit your file at the appropriate link on
Brightspace.
• If you have changes after submission, you can resubmit before the deadline.
Only the last submitted file before the deadline will be marked.
• In the next page, you must certified that this is your own work by signing at
the different places.
• You .pdf must be legible. The questions must be in the right order and the files
should have the co
ect orientation
• Your file must be the following format: LastName,FirstName/Name of file. Fo
example, if my name is Matt Lemire, I would name my file as:
Lemire, Matt Assignment 2.pdf
• Show your work: Means that you must show all your steps with justification.
• No decimal answers will be accepted. We only want exact and simplified answers
in the form of fractions. For example, 0.125 is not accepted but
1
8
would be.
An expression of the form
3π − 1
2
would an example of an exact answer.
• You can use the Discord forum to write privately to other people in the class
egarding answers and work but please do not post any kind of solutions o
major hints on the forum. The goal is for you to learn as much as you can
your own. It is okay to get help from others as long you understand it yourself
eventually.
Question 0a. This assignment is open book. I would kindly ask you to do this
assignment without just copying down other people answers. I would kindly ask you
to promise (code of honour) that you accept the following: I promise not to have
someone else doing my assignment. I am allowed to consult textbooks, notes, the
internet, some classmates, but I will only do so to help my understanding and not fo
others to do my work.
Signature:
(For students who do not write on a printed version of the exam, simply write 0a:
and then put your signature.)
Question 0b: By signing here, I hereby certify that I have read all the instruc-
tions and conditions on the first page and that I will follow them.
Signature:
(For students who do not write on a printed version of the exam, simply write 0b:
and then put your signature.)
Question 0c: By signing here, I hereby certify that I understand that I must submit
all my work no later than Friday Fe
uary 4th, no later than 22:00 at the appropriate
link on Brightspace. I also know that my work will not be accepted passed that day
and time.
Signature:
(For students who do not write on a printed version of the exam, simply write 0c:
and then put your signature.)
Important Trigonometric Identities:
sin2(x) + cos2(x) = 1 cos2(x) =
1 + cos(2x)
2
sin2(x) =
1− cos(2x)
2
sin(x) sin(y) =
1
2
cos(x−y)− 1
2
cos(x+y) =
1
2
cos(y−x)− 1
2
cos(x+y) = sin(y) sin(x)
cos(x) cos(y) =
1
2
cos(x−y)+ 1
2
cos(x+y) =
1
2
cos(y−x)+ 1
2
cos(x+y) = cos(y) sin(x)
sin(2x) = 2 sin(x)cos(x) cos(2x) = cos2(x)− sin2(x)
1. Evaluate
∫∫
R
(√
x
y2
)
, where R : 0 ≤ x ≤ 4, 1 ≤ y ≤ 2. Show all your work.
2. Find the volume of the solid bounded above by z = f(x, y) = 16 − x2 − y2 and
elow by the square S : 0 ≤ x ≤ 2, 0 ≤ y ≤ 2. Show all your work.
3. Evaluate
∫∫
R
xy dA, where R is the region bounded by y =
√
1− x2, x ≥ 0 and
y = 0. Show all your work.
4. Evaluate
∫∫
R
(x4 + y2) dA, where R is the region bounded between y = x3 and
y = x2. Show all your work.
5. Evaluate
∫∫
R
e−y
2/2 dA, where R is the triangle formed by the y-axis, the line
x = 2y and y = 1 Show all your work.
6. Consider a function of two variables f(x, y) (it does not really matter what the
actual function is) and the double integral
∫ 3
0
∫ √9−x2
−
√
9−x2
f(x, y) dy dx. If we change
the order of integration of this double integral, then we obtain
∫
a
∫ d
c
f(x, y) dx dy.
What should a, b, c and d be?
7. Compute the double integral
∫ 1
0
∫ 1
y
cos(x2) dxdy. Show your work.
Hint: I would not waist energy trying to find
∫
cos(x2)dx if I were you...
8. The following is a very interesting fact about double integrals: If R is a region
of R2 of type 1 or of type 2, then the area of this region is given by
∫∫
R
1 dA =
∫∫
R
dA.
Use this (and only this fact) to find the area of the region in R2 that is between
the parabola y = −x2 + 1 and the parabola y = x2 − 17. Show all your work.
9. Evaluate
∫∫
R
y2
x2 + y2
dA, where R is the region between the circles x2 + y2 = 1
and x2 + y2 = 9. Show all your work.
10. Use a double integral over polar coordinates (and only that method) to find the
volume of the solid satisfying the equations x2 + y2 ≤ 1, y ≥ 0, under the paraboloid
z = 2− x2 − y2 and above the xy-plane. Show all your work.