MTH 212 – Kilner
Written Assignment 3
Summer 2025 – Session 2
Be sure to read all instructions on this coversheet before
completing and submitting your assignment.
Failure to follow any of these directions may result in a penalty.
• Solve each of the given problems on separate paper or electronically using a tablet.
• For all problems, you must provide sufficient work to support your final answers.
• You will be graded based on the correctness and completeness of your work,
including the correct and appropriate use of notation (don’t turn in “scrap work”).
• Only methods covered in this course may be used.
• Your final work must be neat and organized (in the order of the problems as they
appear on this page).
• While students are permitted to discuss the problems on this assignment, the work
that you submit must be your own. Students submitting nearly identical work, may
be penalized if it is determined that the solutions were not written up
independently.
• Posting problems from this assignment to any website and/or obtaining solutions
from the web is strictly prohibited. Anyone found violating this rule will receive a 0
on the assignment and will be reported to the appropriate offices at MCC.
Your assignment must be submitted as a SINGLE PDF in
Brightspace in the same location where you accessed it.
Note: Genius Scan and CamScanner are two free phone apps that are easy to use to
take pictures of your work and convert them to a single PDF.
*Due by the end of the day on Tuesday, August 5*
*Make sure your name is on the first page of your assignment.
(This assignment potentially covers material from Units 3.1 – 3.5)
[A] Create a detailed solution for the evaluation of the following double-iterated integral. Your
solution must contain correct notation and should not skip any of the main steps of the
calculation (you can combine basic numerical calculation steps though).
1
𝑒𝑦
𝑦
𝑑𝑥 𝑑𝑦
𝑒 −𝑦 𝑥
∫ ∫
0
[B] Rewrite the following double-iterated integral by changing the order of integration. Note,
because the function 𝑓(𝑥, 𝑦) isn’t specified, you will not be evaluating the integral. *Include a
sketch of the region of integration. Your sketch must include numerical labels on the 𝑥- and 𝑦axes to indicate the scale. Place an ℛ inside the region of integration to make sure I can clearly
identify what the region is.
7
√𝑥−3
∫ ∫
3
𝑓(𝑥, 𝑦) 𝑑𝑦 𝑑𝑥
(𝑥−3)/2
[C] Consider the following iterated integral.
0
√4−𝑦 2
𝑥𝑦
] 𝑑𝑥 𝑑𝑦
2
2
−2 −√4−𝑦2 3 − cos(𝑥 + 𝑦 )
∫ ∫
[
(C.1) Sketch the region of integration.
(C.2) Rewrite the integral in the order 𝑑𝑦 𝑑𝑥. Do NOT evaluate it.
(C.3) Rewrite the integral in polar coordinates. Do NOT evaluate it, but simplify the integrand
the results as much as possible (but no antiderivatives).
There’s more on the next page…
[D] Let 𝑄 be the tetrahedron (shown in the figure below) with vertices (0, 0, 0), (1, 1, 0), (0, 1, 0),
and (0, 1, 1).
The slanted top of this tetrahedron is a portion of the plane given by 𝑧 = 𝑦 − 𝑥.
Consider the triple integral
∭ 𝐹(𝑥, 𝑦, 𝑧) 𝑑𝑉
𝑄
In (D.1) – (D.3) rewrite the integral as iterated integrals in the indicated order (do not evaluate
them, which you can’t since 𝐹 is not specified).
(D.1) The order 𝑑𝑥 𝑑𝑦 𝑑𝑧
(D.2) The order 𝑑𝑦 𝑑𝑧 𝑑𝑥
(D.3) The order 𝑑𝑧 𝑑𝑥 𝑑𝑦
[E] Let 𝐶 be the curve parameterized by: 𝑥 = 𝑡, 𝑦 = 𝑡 3 for 0 ≤ 𝑡 ≤ 1. Evaluate the following line
integral. If done correctly, things should simplify “nicely”.
∫ √1 + 9𝑥𝑦 𝑑𝑠
𝐶