MAT136 – Integral Calculus - Winter 2025 Written Assignment
MAT136 Assignment 3 (Written Component)
Due Monday 10 February 2025, at 9pm
Submitting the Assignment
Due date: Monday, 10 February 2025, at 9pm.
Where to submit: On Crowdmark. Check your email for an invitation to Crowdmark.
What to do: Write solutions to the problems on blank or lined sheets of paper, or write them digitally. Start each
question on a new page. Then photograph, scan, or export a PDF of your solutions. See the Written Assignments
page on Quercus for details.
Uploading: Click on the link in the email from Crowdmark to submit your solutions. Plan on giving yourself some
extra time to upload your solutions the first time.
More details
-1. Changelog: 4 Feb - 4:15pm: Fixed a typo in the setup of Q1. The intersection point should be (4, 2), not (2, 4)
0. Wait, it’s due on Monday? Yes. We decided to extend the assignment by two days.
1. Deadline: The deadline to submit this assignment is strict, to the second. Assignments that are even a few seconds
late will normally receive a grade of 0. Technical issues are not a valid reason to be late, so you are taking a risk if you
leave uploading your solutions to the last minute. Please give yourself lots of time to upload your assignment!
2. Purpose and feedback: The purpose of the written assignments is to give you some practice in thinking about and
writing solutions to mathematical problems, without any time pressure. You will receive feedback on your writing and
on your solutions. You are encouraged to take this opportunity to carefully write your solutions and think about how
to best present your reasoning behind them.
3. Writing solutions: Explain all your work, show your steps as well as your reasoning. You should write in words what
you are doing and why. The person reading your solution should easily be able to understand what you have done,
because you explained it, in words. Submissions with little or no written explanations will not receive full marks.
4. Uploading: A handwritten assignment can be photographed or scanned. It is important that the images (the scans
or photos) are clear and easy to read (not too dark, too bright, too blurry, etc.). Your submitted files are what will be
marked, so if the grader(s) can’t read them or can’t make sense of what you’ve written, you will not get full marks.
You may also write your solutions on a tablet, or type your solutions (LaTeX only please) as long as your solutions are
clear, easy to read and follow, and are all your own work.
Make sure you upload solutions to each problem to the correct place. If you upload solutions to the wrong problems,
or upload all your solutions to the sameproblem, itmay result in getting 0 on all problemsuploaded to thewrong places.
5. Grading: The assignment is out of 10 marks, with the marks for each question indicated beside it. Marks for each
solution will depend on your answer, as well as the quality of your explanation of those answers.
6. Academic Integrity: The solutions that you submit to this assignment must be all your own work. By submitting this
assignment you declare that:
(a) Your solutions are all your own work, explained in your own words.
(b) You have not copied any part of the assignment solutions from anyone or anywhere.
(c) You have not let anyone else copy any part of your solutions to the assignment.
(d) You have not used Generative AI (e.g., ChatGPT) for any part of the assignment.
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MAT136 – Integral Calculus - Winter 2025 Written Assignment
Background
As you learned on Term Test 1, astronomy students from UofT’s St. George campus (UTSG) recently observed an unidenti-
fied object moving straight down toward them. The object descended quickly, stopping and hovering in the sky above the
McLennan Physical Laboratories.
The object is difficult to see clearly, even as it hovers apparently motionless in the sky. Despite observing the object with their
best telescopes and having their most sensitive radio receivers trained on it, they are only able to get a few scattered data
points about the shape of the object, and some scrambled transmissions.
The students are able to get five data points about the top and bottom surfaces of the object, to help them approximate its
shape. They represent these data points as (x, y) coordinates, as follows.
x-coordinate 2 4 6 8 10
y-coordinates of top surface 3 5.5 5 3 2.5
y-coordinates of bottom surface 0 1 2 2 2
A plot of these points can be found below, and you can also use this GeoGebra visualization to look at the data. The blue
points and red points represent the top and bottom surfaces, respectively.
These quantities are all in units of tens of metres. For example, the point (2, 3) on the top surface is 20m to the right and
30m above the origin of this coordinate system.
Other than these data points, the astronomy students are quite unsure of the actual shape of the object. Their best idea
is to use their data to approximate the shape of the region enclosed by the two surfaces, then rotate that region around a
coordinate axis to try to approximate the object as a solid of revolution.
Here’s a plot of the data points from the table above.
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MAT136 – Integral Calculus - Winter 2025 Written Assignment
Problems
1. (3 points) Unfortunately, it’s been a while since these students took MAT136, and they’re rusty. They try to refresh their
memories about solids of revolution, and fall back intomaking common errors while working on the following problem.
Problem
Consider the regionR, in the first quadrant, bounded between the curves y =
√
x, y = x−2, and the x-axis
Find the volume of the solid of revolution obtained by rotatingR around the x-axis.
They are able to find the intersection point of the two curves correctly:
√
x = x− 2 =⇒ x = 4 and therefore y = 2, so the intersection point is (4, 2).
But things do not go well from that point onward.
(a) (1 point) Jimmy is the first student to propose a solution.
Jimmy’s attempt
The
√
x curve is above the x− 2 line between 0 and 4. The region starts at x = 0 and ends at x = 4.
That means its volume is:
V =
∫ 4
0
[
pi
(√
x
)2 − pi(x− 2)2] dx
Jimmy is wrong. In at most a few sentences (and some pictures, if you like), explain the error(s) he is making.
(b) (1 point) Next, Suzie tries the question.
Suzie’s attempt
I can see that this integral should be in two parts, changing at x = 2. The cross-sections of the first
part are circles with radius
√
x, and the cross-sections of the second part are washers with inner radius
x− 2 and outer radius√x, respectively. That means the volume is:
V =
∫ 2
0
(
pi
√
x
)2
dx+
∫ 4
2
pi
(√
x− (x− 2))2 dx
Suzie is alsowrong. In atmost a few sentences (and some pictures, if you like), explain the error(s) she ismaking.
(c) (1 point) Show these students who’s boss by doing a scarier looking version of the same problem.
LetR be the same region described in the problem above (remember,R is in the first quadrant only, above the
x-axis). Find a sum of integrals representing the volume of the solid of revolution obtained by rotatingR around
the the line y = −136.
Explain how you get your final answer, but you do not have to evaluate any integrals.
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MAT136 – Integral Calculus - Winter 2025 Written Assignment
Having proven their skills, UTM MAT136 students are called in to help the UTSG astronomers understand the mysterious
object, which they hope is an alien spacecraft. We have the data from the table, but we can’t know its true shape for sure. So,
the UTM MAT136 students do what they know best: approximate the shape with rectangles.
Over each of the four intervals [2, 4], [4, 6], [6, 8], and [8, 10], use the y-values from the two surfaces at the left endpoint of
the interval as the top and bottom of a rectangle. For example, over the interval [4, 6], the rectangle extends from 1 to 5.5.
Doing this for all four intervals produces a region S , consisting of four side-by-side rectangles.
2. (2 points) The UTM students start by doing the simplest things they can with the region S.
(a) (1 point) Draw a picture of S , and find the area of S.
(Youmay drawon the plot or print a copy of it to drawon. No integration required for the area…it’s just rectangles.)
(b) (1 point) Our first attempt at approximating the mysterious object is as the solid of revolution formed by rotating
S around the x-axis. Find the volume of this solid.
The UTSG students are impressed. However, they note that the solid from Q2 doesn’t look much like the cool flying saucer
they hoped they had discovered, and you agree.
3. (2 points) Find the volume of the solid of revolution obtained by rotating the region S around the y-axis. (This looks
much more like a spaceship!) Be sure to explain your process for doing this computation in detail.
Eventually, they are able to see the precise outline of the re-
gion U they were approximating with S earlier. See the im-
age to the right.
They find that U extends all the way to the y-axis with two
horizontal lines, as shown. They also find that y = g(x) is
a straight line joining (2, 3) to (4, 6), and that the bottom
surface is given by the graph of the function
f(x) =
2
ln(9) ln(x− 1)
for 2 ≤ x ≤ 10.
The curvex = h(y), for 2 ≤ y ≤ 6, forming part of the top surface is an alien curve that the UTM students do not recognize.
The can only unscramble part of one signal from the ship, which says:
...tra! I’m back ... kill ... all ... human ... And by the way,
∫ 6
2
[
h(y)
]2
dy = ...
The students are a bit worried about the first part, but also frustrated that they couldn’t unscramble the RHS of the equation.
They decide to call the missing value of the integral in the messageA.
4. (3 points) Using the information they have gathered above, find the volume of the ship, which is the solid obtained by
rotating the image in the picture about the y-axis.
When simplified, the final answer should involve the constant A, a logarithm, a pi, and some integers. It should not
contain integrals. Be sure to lay out your work clearly, and explain your steps in words and/or pictures.
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