Ch8MoreSec2Sol
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Math 116 / Exam 1 (February 8, 2016)
DO NOT WRITE YOUR NAME ON THIS PAGE
page 9
7
. [10points] MaizeandBlueJewelryCompanyistryingtodecideonadesignfortheirsignature
aMaize-ing bracelet. There are two possible designs: type
W
and type
J
. The company has
done research and the two bracelet designs are equally pleasing to customers. The design for
both rings starts with the function
C
(
x
)=cos
(
π
2
x
)
where all units are in millimeters. Let
R
be the region enclosed by the graph of
C
(
x
) and the graph of
-
C
(
x
) for
-
1
≤
x
≤
1.
a
. [5points] Thetype
W
braceletisintheshapeofthesolidformedbyrotating
R
aroundthe
line
x
=50. Writeanintegralthatgivesthevolumeofthetype
W
bracelet. Include
units
.
Solution:
The volume of the type
W
bracelet, in mm
3
, using the shell method, is
integraldisplay
1
-
1
2
π
(50
-
x
)
·
2
C
(
x
)
dx.
b
. [5 points] The type
J
bracelet is in the shape of the solid formed by rotating
R
around
the line
y
=
-
50. Write an integral that gives the volume of the type
J
bracelet. Include
units
.
Solution:
The volume of the type
J
bracelet, in mm
3
, using the washer method, is
integraldisplay
1
-
1
π
(50+
C
(
x
))
2
-
π
(50
-
C
(
x
))
2
dx.
University of Michigan Department of Mathematics
Winter, 2016 Math 116 Exam 1 Problem 7 (jewelry) Solution
Math 116 / Exam 1 (February 11, 2019)
page 7
6
. [12 points] Ryan Rabbitt is making a smoothie with his new electric drink mixer. Mathemat-
ically, the container of the mixer has a shape that can be modeled as the surface obtained by
rotating the region in the first quadrant bounded by the curves
y
= 27 and
y
=
x
3
/
2
about the
y
-axis, where all lengths are measured in centimeters.
a
. [7 points] Write, but do not evaluate, two integrals representing the total volume, in cm
3
,
the mixer can hold: one with respect to
x
, and one with respect to
y
.
Answer
(with respect to
x
):
integraldisplay
9
0
2
πx
parenleftBig
27
-
x
3
/
2
parenrightBig
dx
Answer
(with respect to
y
):
integraldisplay
27
0
π
parenleftBig
y
2
/
3
parenrightBig
2
dy
b
. [5 points] Ryan adds 1600 cubic centimeters of liquid to his mixer. The container spins
around the
y
-axis at a very high speed, causing the liquid to move away from the center
of the container. The result is the solid made by rotating the shaded region around the
y
-axis in the diagram below. Note that this means that there is an empty space inside
the liquid that has the shape of a cylinder.
r
y
= 27
y
=
x
3
/
2
x
y
Let
r
be the radius of this cylinder of empty space. Set up an equation involving one or
more integrals that you would use to solve to find the value of
r
.
Do not
solve for
r
.
Solution:
integraldisplay
9
r
2
πx
parenleftBig
27
-
x
3
/
2
parenrightBig
dx
= 1600
,
or
integraldisplay
27
r
3
/
2
π
parenleftBig
y
2
/
3
parenrightBig
2
dy
-
πr
2
(27
-
r
3
/
2
) = 1600
.
(There are other equations that would also work.)
Answer:
University of Michigan Department of Mathematics
Winter, 2019 Math 116 Exam 1 Problem 6 (smoothies) Solution
Math 116 / Exam 1 (February 9, 2015)
page 4
2
. [13 points]
Fred is designing a plastic bowl for his dog, Fido.
Fred makes the bowl in the
shape of a solid formed by rotating a region in the
xy
-plane around the
y
-axis. The region,
shaded in the figure below, is bounded by the
x
-axis, the
y
-axis, the line
y
= 1 for 0
≤
x
≤
4,
and the curve
y
=
-
(
x
-
5)
4
+ 2 for 4
≤
x
≤
2
1
/
4
+ 5. Assume the units of
x
and
y
are inches.
x
y
a
. [7 points]
Write an expression involving one or more integrals which gives the volume of
plastic needed to make Fido’s bowl. What are the units of your expression?
Solution:
Using the cylindrical shell method, we have that the volume of plastic needed
to make Fido’s bowl is given by
integraldisplay
4
0
2
πxdx
+
integraldisplay
5+2
1
4
4
2
πx
(2
-
(
x
-
5)
4
)
dx
.
Using the washer method, we have that the volume of plastic needed to make Fido’s
bowl is given by
π
integraldisplay
1
0
(5 + (2
-
y
)
1
/
4
)
2
dy
+
π
integraldisplay
2
1
(5 + (2
-
y
)
1
/
4
)
2
-
(5
-
(2
-
y
)
1
/
4
)
2
dy.
The units for either expression are in
3
.
b
. [6 points]
Fred wants to wrap a ribbon around the bowl before he gives it to Fido as a
gift. The figure below depicts the cross section of the bowl obtained by cutting it in half
across its diameter. The thick solid curve is the ribbon running around this cross section,
and the dotted curve is the outline of the cross section which is not in contact with the
ribbon. Write an expression involving one or more integrals which gives the length of the
thick solid curve in the figure (the length of ribbon Fred needs to wrap the bowl).
Solution:
The length of ribbon Fred needs to wrap the bowl is given by
10 + 2(2
1
4
+ 5) + 2
integraldisplay
5+2
1
4
5
radicalbig
1 + 16(
x
-
5)
6
dx
.
x
y
University of Michigan Department of Mathematics
Winter, 2015 Math 116 Exam 1 Problem 2 (dog bowl) Solution
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Math 116 / Exam 1 (October 8, 2014)
page 8
7
. [13 points] Kazilla is designing a new board game. She is interested in using the region
R
in
the
xy
-plane bounded by
y
= 2,
y
=
x
,
x
= 1 and
x
= 0.
a
. [4 points] The first part of the game is a spinning top formed by rotating the region
R
around the
y
-axis. Write an integral (or a sum of integrals) that gives the volume of the
spinning top. Do not evaluate your integral(s).
Solution:
Shell method:
integraldisplay
1
0
2
π
(2
-
x
)
xdx
Washer method:
integraldisplay
1
0
πy
2
dy
+
integraldisplay
2
1
πdy
b
. [4 points] Another game piece has a base in the shape
R
, but with semicircular cross
sections
perpendicular
to the
x
-axis. Write an integral which gives the volume of the
game piece. Do not evaluate your integral.
Solution:
π
8
integraldisplay
1
0
(2
-
x
)
2
dx
c
. [5 points] A third game piece has volume given by
integraldisplay
2
0
π
(
h
(
x
))
2
dx
where
h
(
x
) is a contin-
uous function of
x
. Use MID(3) to approximate the volume of this third game piece. Be
sure to write out all of the terms in your approximation. Your answer may contain the
function
h
(
x
).
Solution:
MID
(3) =
2
3
[
π
(
h
(1
/
3))
2
+
π
(
h
(1))
2
+
π
(
h
(5
/
3))
2
]
University of Michigan Department of Mathematics
Fall, 2014 Math 116 Exam 1 Problem 7 (board game) Solution