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Problem 7.13
For the area shown, use integration to determine the
x
and
y
positions of the
centroid.
(a) Use a vertical area element.
(b) Use a horizontal area element.
Solution
The yposition of the centroid is given by
02x x21
22x x2dx
3x4
2Cx5
10 ˇˇˇ
2mm
0
Part (b)
For a horizontal area element, we take the independent variable to
be
y
, as shown at the right. The area element is
dA Dw dy
, and to determine
w
, we use the quadratic formula to solve the equation
yD2x x2
to obtain
the two values of x, namely xrand xl, that correspond to a specific value of y.
Statics 2e 987
The xposition of the centroid is given by
0.1/2p1y dy
which is simplified to find
Problem 7.14
For the area shown, use integration to determine the
x
and
y
positions of the
centroid.
Solution
For the area of this problem, a vertical area element is more convenient than a horizontal area element. Thus,
with ytD1Cx=2 and ybDx2, the area element dA, and its centroid Qxand Qyare
The xlocation of the centroid is given by
0x1Cx
2x2dx
2Cx3
6x4
4ˇˇˇ
1in:
0
The ylocation of the centroid is given by
0
1
21Cx
2Cx21Cx
2x2dx
1
2xCx2
2Cx3
12 x5
5ˇˇˇ
1in:
0
Problem 7.15
For the area shown, use integration to determine the
x
and
y
positions of the
centroid.
Solution
A vertical area element will be used. Thus, with
ytDpr2x2
and
ybD0
, the area element
dA
, and its
centroid Qxand Qyare
The xlocation of the centroid is given by
rxpr2x2dx
pr2x2r2
3Cx2
3ˇˇˇ
r
r
Problem 7.16
For the area shown, use integration to determine the
x
and
y
positions of the
centroid.
Solution
For the area in this problem, a vertical area element is more convenient than a
horizontal area element. The two lines intersect at xwhere
Statics 2e 991
The xlocation of the centroid is given by
0QxdA CR12 in.
6:4 in. QxdA
The ylocation of the centroid is given by
0QydA CR12 in.
6:4 in. QydA
Problem 7.17
For the area shown, use integration to determine the
x
and
y
positions of the
centroid.
2.ytCyb/D1
2xx2
40 :(6)
For the region 15 mm x20 mm,
Statics 2e 993
The xlocation of the centroid is given by
0QxdA CR20 mm
15 mm QxdA
The ylocation of the centroid is given by
0QydA CR20 mm
15 mm QydA
Problem 7.18
Determine expressions for lines
y1
and
y2
, and then use integration to determine
the xand ypositions of the centroid.
Solution
We begin by determining the equations for the bottom boundary
.y1/
and top boundary
.y2/
of the area, as
follows
The xlocation of the centroid is given by
0x20 in:Cx
2Cx
4dx
4ˇˇˇ
80 in:
0
Problem 7.19
Determine expressions for lines
y1
and
y2
, and then use integration to determine
the xand ypositions of the centroid.
Solution
We begin by determining the equations for the bottom boundary
.y1/
and top boundary
.y2/
of the area, as
follows
The xlocation of the centroid is given by
0x5mx
2Cx
8dx
2x3
8ˇˇˇ
8m
0
Problem 7.20
Determine expressions for lines
y1
and
y2
, and then use integration to determine the
x
and ypositions of the centroid.
2.xrCxl/D1
2y
4C23 Cy
2D1
223 C3y
4;(5)
QyDy: (6)
The xposition of the centroid is given by
0
1
223 C3y
423 y
4dy
Problem 7.21
For the area in the figure cited below, use composite shapes to determine the
x
and
y
positions of the
centroid.
Figure P7.18.
Solution
To determine the location of the centroid, we will use the three composite shapes shown.
D.10 in./.1600 in.2/C.33:33 in./.1600 in.2/C.6:667 in./.800 in.2/
Problem 7.22
For the area in the figure cited below, use composite shapes to determine the
x
and
y
positions of the
centroid.
Figure P7.19.
Solution
To determine the location of the centroid, we will use the three composite shapes shown.
Using the information in the table, the locations of the centroid are
NyDPQyiAi
PAi
(1)
Problem 7.23
For the area in the figure cited below, use composite shapes to determine the
x
and
y
positions of the
centroid.
Figure P7.20.
Solution
To determine the location of the centroid, we will use the three composite shapes shown.
Using the information in the table, the location of the centroid is
NyDPQyiAi
PAi
(1)
In summary, the centroid is located at
Problem 7.24
For the triangle shown, having base
b
and height
h
, use integration to show that the
y
position of the centroid is
NyDh=3
.Hint: Begin by writing an expression for
the width of a horizontal area element as a function of y.
Problem 7.25
Use integration to determine the
x
and
y
positions of the centroid. Express your
answers in terms of rand ˛.
QxD2
Problem 7.26
Determine constants
c1
and
c2
so that the curves intersect at
xDa
and
yDb
.
Use integration to determine the
x
and
y
positions of the centroid. Express your
answers in terms of aand b.
a3.(1)
Using these results for c1and c2, the equations for the top and bottom boundaries of the area are
ytDb
papxand ybDb
2.ytCyb/D1
2b
a3x3:(5)
The xlocation of the centroid of the area is given by
0xb
papxb
a3x3dx
5a3C2bx2
5qx
aˇˇˇ
a
0
Problem 7.27
For the straight line shown, set up the integrals, including the limits of inte-
gration, that will yield the length of the line, and the
x
and
y
positions of the
centroid. Evaluate these integrals.
Problem 7.28
For the straight line shown, set up the integrals, including the limits of inte-
gration, that will yield the length of the line, and the
x
and
y
positions of the
centroid. Evaluate these integrals.
Problem 7.29
For the line shown:
(a)
Set up the integrals for integration with respect to
x
, including the limits
of integration, that will yield the xand ypositions of the centroid.
(b) Repeat Part (a) for integrations with respect to y.
(c)
Evaluate the integrals in Parts (a) and/or (b) by using computer software
such as Mathematica or Maple.
