MAT 142 – Calculus with Analytic Geometry II
Take-Home Exam 3 – Applications of Integration
Due Monday, April 3rd in class
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1.
INSTRUCTIONS
Show your work, neatly, on separate sheets of paper for all the following
problems to receive full credit.
Leave your answers in exact form whenever possible.
Justify all your results analytically.
Provide sketches or computer-generated images when appropriate.
Find the area of the region bounded by the following curves. Explain your integral of
integration.
a) x  y  0 , x  y 2  3 y .
b) y 
x3
8x
, y 2 .
2
x 1
x 1
2.
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Use a calculus-based slicing method to find the volume of the Great Pyramid in Giza,
Egypt.Use the dimensions of this square-based pyramid provided in the figure below.
3.
Let  be the region in the first quadrant bounded by the curves y  x3 and y  2 x  x 2 .
a) Compare the volumes of the two solids obtained by rotating  about the x -axis
and the y -axis. Which is the biggest?
b) Find the volume of the solid obtained by rotating  about the line y  1 .
4.
Exercise # 57 on p. 436
5.
Each of the integrals below corresponds to the volume of a solid of revolution.
Describe precisely how these solids of revolution are generated from a rotated region.
[Note: there may be multiple valid answers here.]
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  2  cos y  dy
2
2
a)
2
0
6.
b)
 2  x cos x dx
2
0
1
c)    x 4  4 x 2  4 x  x  dx
0
A hemispherical bowl of radius 10 inches is filled with water to a depth of h inches (the
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bowl is empty when h  0).
a) Use a method of your choice to find the volume of water in the bowl, in cubic
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inches, as a function of the depth h . Check the special cases h  0 and h  8 .
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b) To what depth should this bowl be filled to have exactly 100 cubic inches of water?
Round your answer to the nearest hundredth of an inch.
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7.
Use the method of cylindrical shells to find the volume of a torus (doughnut) formed
when the circle of radius 4 centered at (0, 6) is revolved about the x -axis.
8.
Which of the following arc lengths is greatest: x  e y for 0  y 
0 x
9.
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4
?
Consider the function f (x)  ax(1 x) on the interval 0,1, where a is any positive
real number. Find the point(s) at which the value of f equals its average value. Are
these points dependent of a ?
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10.

1
or y  ln  sec x  for
2
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1 
, the x -axis and the line x  2 . Let S the
x ln x
solid of revolution obtained from rotating R about the x -axis.
 bounded by y 
Let R be the region
 what is it?
Is the volume of S finite or infinite? If finite,
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