MAT1322
Calculus II
Midterm 1
Tuesday, May 30, 2023
Instructions:
• Detailed instructions for the format and technical requirements of Test 1 have been announced in
Brightspace.
• Duration of Test-writing Phase: 60 minutes
• This test consists of 3 multiple-choice questions, one Bonus multiple-choice question, and 4 shortor long-answer questions worth a total of 30 points.
• All your answers must be written on your own blank paper (lined paper or graph paper is fine).
You will need to scan and upload a copy of your work before the test deadline. On each page you
plan to submit, write your name at the top.
• MULTIPLE-CHOICE QUESTIONS
Questions 1–3 and the bonus question are multiple-choice questions worth a total of 7 points.
Your answers to multiple-choice questions do not need to be justified. You may write your scrap
work on your paper but it will not be graded. When you reach your answer, clearly indicate the
question number and write the letter of your response beside the question number:
For example: (write out your scrap work, but it will not be graded)
(clearly indicate your final choice)
Q1. [letter of your choice]
• SHORT- AND LONG-ANSWER QUESTIONS
Questions 4–7 are short- or long-answer questions worth a total of 23 points. For these
questions, unless specified otherwise, all of your work must be justified and your steps must be
written in a clear and logical order. Clearly indicate Question numbers.
For example:
Q5(b). [write a fully justified solution].
• STUDENT DECLARATION
Within your scanned pdf document, you must include a hand-written declaration, to indicate that
you will honour the terms of the test, as follows:
‘‘I,
[your name printed clearly] , promise to uphold my academic integrity.
I wrote my solutions without help from any unauthorized resource or person.
• Below your statement, you must include your signature and the date.
• Scan-and-Upload Phase: When the test-writing phase ends, you will be given 10 minutes to
scan and create a pdf copy of your work which you will upload and submit in the Brightspace
Assignment for Test 1. Your scan should include your Cheat Sheet and your student declaration.
The file name should include your full name.
•
Good luck!
1
MULTIPLE-CHOICE QUESTIONS
Q1. [2 points] Which of the following integrals computes the arc length of the curve y = 2x3 − x on
the interval [0, 1]?
Z 1√
A.
2x3 − x + 1 dx
Z
B.
0
Z
√
1
3
2
0
Z
4x6 − 4x4 + x2 + 1 dx
C.
√
36x4 − 12x2 + 2 dx
0
1
6
4
2
2
(4x − 4x + x + 1) dx
E.
1
Z
0
(2x − x + 1) dx
D.
1
0
Z
F.
1
(36x4 − 12x2 + 2)2 dx
0
Q2. [2 points] Consider a reservoir of water 8m tall. It is filled up to 6m high with water. At the base
of the tank, there is a door in the shape of a quarter-circle of radius 2m. Which of the following
integrals represents the total hydrostatic force (in Newtons) exerted on the door? Note that the
density of water is 1000 kg/m3 and the acceleration due to gravity is 9.8 N/kg.
Z 2
Z 6
A. 9800
(6 − x) cos(x) dx
B. 9800
(6 − x) cos(x) dx
0
0
2
Z
C. 9800
Z 6
√
√
2
(6 − x) 4 − x dx D. 9800
(6 − x) 4 − x2 dx
0
0
2
Z
E. 9800
Z 6
√
√
2
(8 − x) 4 − x dx F. 9800
(8 − x) 4 − x2 dx
0
Z
0
2
Z
(8 − x) cos(x) dx
G. 9800
0
6
(8 − x) cos(x) dx
H. 9800
0
Q3. [2 points] Consider a region R in the plane bounded above by y = x + 5 and bounded below by
y = 3 − 3x2 , and is contained between x = −1 and x = 1. Suppose a thin metal plate of uniform
density ρ = 7 is shaped like R; let (x, y) represent its center of mass. Given that the area of R is
6, which of the following expressions represents the value of x?
Z 1
Z
Z
7
7 1 2
7 1
3
2
A.
3x + x + 2x dx
B.
3x + x + 2 dx
C.
(x + 5)2 − (3 − 3x2 )2 dx
12 −1
6 −1
6 −1
Z
Z
Z 1
1 1 3
1 1 2
1
2
3x + x + 2x dx
E.
3x + x + 2 dx
F.
(x + 5)2 − (3 − 3x2 )2 dx
D.
6 −1
6 −1
12 −1
G. None of these options.
Bonus. [1 point] In Q3, which of the expressions represents the value of y?
SHORT- AND LONG-ANSWER QUESTIONS
Q4. [2 points] The temperature (in degrees Celsius) during the course of a day is given by the formula
π
T (x) = 15 + 10 sin(x · 24
), where x is the time in hours. Give a definite integral that computes the
average temperature between 9AM and 9PM.
Q5.
(a) [6 points] For each of the following integrals, determine if they are proper or improper. If an
integral is improper, briefly explain why (one sentence), and write it as a limit (or sum of limits)
of proper integrals.
Z
∞
(i)
e
−∞
−2x2
Z
dx
π/2
(ii)
0
Z
1
dx
cos(x)
−1
Z
(b) [2 points] Determine whether the improper integral
the definition seen in class to explain why.
2
(iii)
2x2
dx
(x + 3)3
∞
cos(x) dx is convergent or divergent. Use
0
Q6. Consider the curves y = −2x(x − 2) and y = x.
(a) [1 point] Find the x-values of the points of intersection of these curves. Draw a diagram to
represent the curves and these x-values. Show your work. (Note: if you can’t find these points, for
the following parts, just represent them by a and b, where a < b.)
(b) [2 points] Compute the area of the region R enclosed by these two curves. Show your work.
(c) [3 points] Consider the solid formed by revolving the region R (of part (b)) around the axis
y = −1. Give a definite integral that computes the volume of this solid by the washer method.
(Do not evaluate the integral.) Show your work.
(d) [3 points] Consider the solid formed by revolving the region R (of part (b)) around the axis x = 0.
Give a definite integral that computes the volume of this solid by the shell method. (Do not
evaluate the integral.) Show your work.
Q7. Consider a tank with the following shape and dimensions.
(a) [3 points] Suppose the tank is filled with water. Give a definite integral that computes the total
work (in Joules) required to lift all of the water to 2m above the top of the tank. (Do not evaluate
the integral.) Show your work. Note that the density of water is 1000kg/m3 and the acceleration
due to gravity is 9.8m/s2 .
(b) [1 point] What would the definite integral be if the tank were only filled halfway to the top? You
can give the integral without justification.