Problem 7.1 In Active Example 7.1, Suppose that the
triangular area is oriented as shown. Use integration to
determine the xand ycoordinates of its centroid. (Notice
Solution: The height of the vertical strip is h⊲h/b⊳ x so the area
is dA Dhh
bxdx. Use this expression to evaluate Eq. (7.6).
Problem 7.2 In Example 7.2, suppose that the area is
redefined as shown. Determine the xcoordinate of the
centroid.
x
y
Solution: The height of the vertical strip is 1 x2, so the area is
dA D⊲1x2⊳dx. Use this expression to evaluate Eq. (7.6).
The xcoordinate of the centroid is
Problem 7.3 In Example 7.2, suppose that the area is
redefined as shown. Determine the ycoordinate of the
centroid.
y
Solution: The height of the vertical strip is 1 x2, so the area is
dA D⊲1x2⊳dx.
The ycoordinate of the midpoint of the vertical strip is
11
Problem 7.4 Determine the centroid of the area.
y
Solution: The height of a vertical strip of width dx is x2xC1,
so the area is dA D⊲x2xC1⊳dx. Use this expression to evaluate
Eq. (7.6).
The xcoordinate of the centroid is
xdA
x⊲x2xC1⊳x
4x3
3Cx2
22
Problem 7.5 Determine the coordinates of the centroid
of the area.
y
Solution: Use a vertical strip – The equation of the line is yD
82x
3
ydx D9
382
3xdx D13
xD5.5
512
Problem 7.6 Determine the xcoordinate of the
centroid of the area and compare your answers to the
values given in Appendix B.
Solution:
ADb
dy dx DcbnC1
Problem 7.7 Determine the ycoordinate of the centroid
of the area and compare your answer to the value given
Solution: See solution to 7.6
Problem 7.8 Suppose that an art student wants to paint
a panel of wood as shown, with the horizontal and
vertical lines passing through the centroid of the painted
area, and asks you to determine the coordinates of the
centroid. What are they?
Solution: The area:
AD1
0
⊲x Cx3⊳dx Dx2
2Cx4
41
0D3
4.
Problem 7.9 Determine the value of the constant cso
that the ycoordinate of the centroid of the area is yD2.
What is the xcoordinate of the centroid?
x
4
2
0
Solution: The height of a vertical strip of width dx is cx2so the
area is dA Dcx2dx.
2
34
2
D2)cD0.376
The xcoordinate of the centroid is
xdA
x⊲cx2⊳dx
44
cD0.376, x D3.21
Problem 7.10 Determine the coordinates of the cent-
roid of the metal plate’s cross-sectional area.
y
x
Solution: Let dA be a vertical strip:
The area dA DydxD41
4x2dx. The curve intersects the xaxis
where 4 1
dA D4
441
4x2dx D2x2x4
12 4
4
To determine y, let yin equation (7.7) be the height of the midpoint
of the vertical strip:
ydA
4
1
241
4x241
4x2dx
21.3D1.6ft.
y
y
x
x
dx
dA
514
c
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Problem 7.11 An architect wants to build a wall with
the profile shown. To estimate the effects of wind loads,
he must determine the wall’s area and the coordinates
of its centroid. What are they?
0
1
2
3
4
204
y = 2 + 0.02x2
6810
y, m
x, m
Solution:
dA DydxD⊲2C0.02x2⊳dx
Y
xD10
0
10
0
dA D10
0
26.67
xD2x2
2C0.02 x4
4
10
0
26.67 m
4
xD5.62 m
0y
2ydx
0
⊲2C0.02x2⊳2dx
yD74.67
Problem 7.12 Determine the coordinates of the cen-
troid of the area.
x
y
y ⫽ ⫺ x2 ⫹ 4x ⫺ 7
1
4
Solution: Use a vertical strip. We first need to find the xintercepts.
Problem 7.13 Determine the coordinates of the cen-
troid of the area.
x
y
y ⫽ 5
y ⫽ ⫺ x2 ⫹ 4x ⫺ 7
1
4
Solution: Use a vertical strip. We first need to find the xintercepts.
516
Problem 7.14 Determine the xcoordinate of the cent-
roid of the area.
x
y
y = x3
y = x
Solution: Work this problem like Example 7.2
2x4
41
0
21
4D
4
xD0.533
y
Problem 7.15 Determine the ycoordinate of the cent-
roid of the area shown in Problem 7.14.
Solution: Solve this problem like example 7.2.
ydA
2⊲x Cx3⊳⊲x x3⊳dx
Problem 7.16 Determine the xcomponent of the cent-
roid of the area.
y
Solution: The value of the function yDx2xC1atxD0is
yD1, and its value at xD2isyD3. We need a function describing
a straight line that passes through those points. Let yDax Cb. Deter-
dA D2
0
⊲2xx3⊳dx D2x3
2x4
42
0
Problem 7.17 Determine the xcoordinate of the cent-
roid of the area.
x
y
y = x
y = x2 – 20
Solution: The intercept of the straight line with the parabola
occurs at the roots of the simultaneous equations: yDx, and yDx2
20. This is equivalent to the solution of the quadratic x2x20 D0,
x1D4, and x2D5. These establish the limits on the integration.
The area: Choose a vertical strip dx wide. The length of the strip
is ⊲x x2C20⊳, which is the distance between the straight line
xD60.75
518
Problem 7.18 Determine the ycoordinate of the cent-
roid of the area in Problem 7.17.
Solution: Use the results of the solution to Problem 7.17 in the
following.
121.5D7.6
Problem 7.19 What is the xcoordinate of the centroid
of the area?
y
y ⫽ ⫺ x2 ⫹ 2x
2
1
6
Problem 7.20 What is the ycoordinate of the centroid
of the area in Problem 7.19?
Solution: Use vertical strips, do an integral for the parabola then
subtract the square
Problem 7.21 An agronomist wants to measure the
rainfall at the centroid of a plowed field between two
roads. What are the coordinates of the point where the
rain gauge should be placed?
0.5 mi
y
Similarly:
2C0.3x1.1
0.5D0.2538 sq mile.
The y-coordinate: The y-coordinate of the centroid of the elemental
0.5
D1.1
⊲0.0473x2C0.1154xC0.045⊳dx
0.2538 D0.3995 mi
520
c
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Problem 7.22 The cross section of an earth-fill dam is
shown. Determine the coefficients aand bso that the y
coordinate of the centroid of the cross section is 10 m.
x
y
100 m
y = ax – bx3
Solution: The area: The elemental area is a vertical strip of length
yD3.810a2ð104
Problem 7.23 The Supermarine Spitfire used by Great
Britain in World War II had a wing with an elliptical
profile. Determine the coordinates of its centroid.
y
2b
x2
—
a2
y2
—
b2
+= 1
x
Solution:
y
x2
—
a2
y2
—
b2
+= 1
By symmetry, the xcentroid of the wing is the same as the xcentroid
of the upper half of the wing. Thus, we can avoid dealing with š
values for y.
b
y
dA = y dx
b
a
y = a2 – x2
2Ca2
2sin1x
a
Substituting, we get
xDa2x23/2/3a
0
xpa2x2
2Ca2
2sin1x
aa
522
Problem 7.24 Determine the coordinates of the
centroid of the area.
Strategy: Write the equation for the circular boundary
in the form yD⊲R2x2⊳1/2and use a vertical “strip”
of width dx as the element of area dA.
y
R
Problem 7.25* If RD6 and bD3, what is the y
coordinate of the centroid of the area?
x
R
b
Solution: We will use polar coordinates. First find the angle ˛
R
b
α
Problem 7.26* What is the xcoordinate of the cen-
troid of the area in Problem 7.25?
Problem 7.27 In Active Example 7.3, suppose that the
area is placed as shown. Let the dimensions RD6 in,
cD14 in, and bD18 in. Use Eq. (7.9) to determine the
xcoordinate of the centroid.
y
x
R
b
c
Solution: Let the semicircular area be area 1, let the rectangular
Problem 7.28 In Example 7.4, suppose that the area is
given a second semicircular cutout as shown. Determine
the xcoordinate of the centroid.
y
Solution: Let the rectangular area without the cutouts be area 1,
524
Problem 7.29 Determine the coordinates of the
centroids.
Solution: Break into a rectangle, a triangle and a circular hole
2⊲8⊳⊲6⊳4[⊲2⊳2]
Problem 7.30 Determine the coordinates of the
centroids.
y
10 in
Solution: The strategy is to find the centroid for the half circle
area, and use the result in the composite algorithm. The area: The
Problem 7.31 Determine the coordinates of the
centroids.
y
0.8 m
Solution: Use a big triangle and a triangular hole
xD
2
31.01
2⊲1.0⊳⊲0.8⊳0.6C2
30.41
2⊲0.4⊳⊲0.8⊳
yD
30.81
2⊲1.0⊳⊲0.8⊳1
30.81
2⊲0.4⊳⊲0.8⊳
yD0.267 m
Problem 7.32 Determine the coordinates of the
centroid.
y
23
Solution: Let the area be divided into parts as shown. The areas
The ycoordinate of the centroid of the composite area is
526
Problem 7.33 Determine the coordinates of the
centroids.
y
mm
400
mm
mm
Solution: Break into 4 pieces (2 rectangles, a quarter circle, and
a triangle)
0.2[⊲0.4⊳⊲0.3⊳]C0.44[0.4]
yD
C0.35⊲0.3⊳⊲0.7⊳C1
30.71
2⊲0.3⊳⊲0.7⊳
Problem 7.34 Determine the coordinates of the cen-
troid.
y
2 ft
2
3
Solution: Let the area be divided into parts as shown. The areas
and the coordinates are
Problem 7.35 Determine the coordinates of the
centroids.
y
x
90 mm
20 mm
10
mm
30 mm
30 mm
20 mm
Solution: Determine this result by breaking the compound object
into parts
20 mm
40 mm
y
A1
=+
A1:A1D⊲30⊳⊲90⊳D2700 mm2
x1D45 mm
y1D15 mm
A2: (sits on top of A1)
A2D⊲40⊳⊲50⊳D2000 mm2
x2D20 mm
y2D30 C25 D55 mm
A3:A3D1
2r2
0D
2⊲20⊳2D628.3mm
2
x3D20 mm
y3D80 mm C4r0
3D88.49 mm
A4:A4D⊲30⊳⊲20⊳Cr2
i
A4D600 C⊲10⊳2D914.2mm
2
x4D20 mm
y4D50 C15 D65 mm
Area (composite)
DA1CA2CA3A4
D4414.2mm
2
For the composite:
4414.2D35.3mm
yDy1A1Cy2A2Cy3A3y4A4
yD146675
4414.2D33.2mm
The value for yis not the same as in the new problem statement. This
value seems correct. (The xvalue checks).
528
c
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Problem 7.36 Determine the coordinates of the
centroids.
10
mm
15
mm
15
mm
10
mm
5 mm 5 mm
15 mm 15 mm
5 mm
15 mm
50 mm
x
y
Solution: Comparison of the solution to Problem 7.29 and our
areas 1, 2, and 3, we see that in order to use the solution of
Problem 7.29, we must set aD25 mm, RD15 mm, and rD5 mm.
xD18.04 mm. The corresponding areas for regions 1, 2, and 3 is
1025 mm2. The centroids of the rectangular areas are at their geometric
centers. By inspection, we how have the following information for the
15 mm
y
1
5
15
Problem 7.37 The dimensions bD42 mm and hD
22 mm. Determine the ycoordinate of the centroid of
the beam’s cross section.
y
x
h
b
200 mm
120 mm
530