Name:
Math 2260 Exam #1
February 3, 2012
Instructions: You are welcome to use one sheet of notes, but no other references or tools are allowed
(no textbooks, no calculators, etc.). This is a 50 minute exam; you may start working at 10:10 am
and must stop at 11:00 am. To receive full credit for a correct answer you must demonstrate how you
arrived at that answer. To receive partial credit for an incorrect answer your work must be clearly
explained.
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1. (12 points) The curves determined by the equations y = x2 and x = y 2 enclose a region in the first
quadrant, as seen below.
(a) What is the area of this region?
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(b) Rotating this region around the x-axis yields a solid. Write down the integrals which will determine
the volume of this solid using (i) the washer method and (ii) the shell method.
(i) Washer method integral:
(ii) Shell method integral:
(c) Compute the volume of this solid by picking either of the integrals you wrote down in part (b)
and evaluating it.
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√
2. (5 points) Between x = 0 and x = π, the curve y = sin x2 and the x-axis enclose a region. What
is the volume of the solid obtained by revolving this region around the y-axis?
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3. (5 points) What is the length of the segment of the curve y =
5
x3/2
2
between x = − 16
9 and x = 0?
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4. (5√points)
Revolving the portion of the graph of the function f (x) = x between the points (0, 0) and
2, 2 around the y-axis yields a type of surface called a paraboloid. What is the surface area of this
paraboloid?
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