Learning Experience for Calculus and Analytical Geometry I
MAC2311
Contributed by Scott Demsky, D.A.
Integration
1. Sketch the parabolic arch bounded by 𝑦 = 9 − 𝑥 2 and the x-axis.
Use a definite integral to calculate the area of this arch.
2. Calculate the base, b, and the height, h, of the arch in part 1 and write a formula for the
area, A, of the arch in the form 𝐴 = 𝑘𝑏ℎ where k is a constant.
3. Prove that the formula, 𝐴 = 𝑘𝑏ℎ, from part 2 is true for any parabolic arch bounded by
𝑦 = 𝑐 − 𝑎𝑥 2 (where c and a are arbitrary positive constants) and the x-axis. This is called
Archimedes’ Formula. Proceed as follows:
a. Calculate the base, b, and the height, h.
b. Use a definite integral to calculate the area of the arch, A, and show that it’s equal
to kbh.
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Intellectual Standards (“CARL”)
“…The ultimate goal of critical thinking is to question answers rather than to
answer questions.” (John D. Eigenauer)
 Clarity
o Is the problem statement clear? Do you understand it?
o Are the statements in your solution clear and understandable?
 Accuracy
o Are the calculations and assertions in your solution correct?
o Is your answer correct?
 Relevance
o What given information is relevant to solving the problem?
o In what part(s) of your solution is the given information needed?
o Which Calculus concepts/facts are relevant to solving the problem?
o Was the information and reasoning that you used necessary to solve
the problem?
 Logic
o
o
o
o
Are the steps in your solution logically correct?
Does each step follow from the previous step(s)?
Are the steps justified by mathematical truths?
Is your mathematical reasoning correct?
In summary, is your solution clear, accurate and relevant to the question asked
and is your mathematical reasoning logically correct?
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Grading Rubric (“CARL”)
Critical Thinking assignments will be graded using the following rubric (100 points
maximum).
21-25 points
16-20 points
11-15 points
6-10 points
0-5 points
Clarity
All statements in
the solution are
understandable.
Most
statements in
the solution
are
understandabl
e.
Some statements
in the solution are
understandable;
some are not.
Many statements in
the solution are not
understandable.
Few, if any, of the
statements in the
solution are
understandable
Accuracy
All calculations
and assertions are
correct.
Most
calculations
and assertions
are correct.
Some calculations
and assertions are
correct; some are
not.
Many calculations or
assertions are
incorrect.
Few, if any, calculations
or assertions are correct.
Relevance
All information
and reasoning
used is necessary
to solve the
problem.
Most
information
and reasoning
used is
necessary to
solve the
problem.
Some information
and reasoning used
is necessary to
solve the problem;
some is not.
Much information or
reasoning used to
solve the problem is
not necessary.
Little or no information
or reasoning used to
solve the problem is
necessary.
Logic
Final answer and
all mathematical
reasoning are
correct.
Final answer is
correct and
most
mathematical
reasoning is
correct.
Some
mathematical
reasoning is
correct; some is
not. Final answer
may or may not be
correct.
Final answer and/or
much mathematical
reasoning are
incorrect.
Final answer and most
mathematical reasoning
are incorrect.
3