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Calculus II
Spring 2021 (April 23)
Midterm Exam
Total 3 problems (50 points)
• At the top of the first page of your answer sheet should contain your name and
ID.
Ex) 김선달 (Seon-Dal Kim), # 20210000
• Write the following honor pledge and sign it.
“I affirm that I will not give or receive any unauthorized help on this exam, and
that all work will be my own. Sign:
”
• Show all your works. Write legibly in a logical and organized fashion.
1. Consider the cycloid given by
x = t − sin t,
y = 1 − cos t.
(a) (6 points) Find the length of one arch of the cycloid
(b) (10 points) Find the area of the surface generated by revolving one arch of
the cycloid about the x-axis. Z
Z
sin2 t cos t 2
3
( If necessary, use the formula sin t dt = −
+
sin t dt. )
3
3
(c) (6 points) Find the volume swept out by revolving the region bounded by the
x-axis and one arch of the cycloid about x-axis.
2. (a) (8 points) Find T and N for
r(t) = ti + t2 j
at t = a.
(b) (10 points) Find the center of the osculating circle for the parabola y = x2
when x = a.
3. (10 points) Show that
f (x, y) =
is continuous in R2 .




4xy 2
, (x, y) 6= (0, 0)
x2 + y 2



0,
(x, y) = (0, 0)