Theorem pappus (1)

9.3 Theorems of Pappus and Guldinus

9.3 Theorems of Pappus and Guldinus Example 1, page 1 of 2
10 ft
y
x
3 ft
3 ft
10 ftC
= 1.5 ft
3 ft
2
x
y
1. Determine the amount of paint required to paint
the inside and outside surfaces of the cone, if one
gallon of paint covers 300 ft2
.
1 The y axis is the axis of revolution.
The generating curve is a
straight line through the
origin.
2
The distance to the centroid of the line is
rC = 1.5 ft
3

3 ft
10 ft
4 The length of the curve can be found from the Pythagorean theorem:
L = (10 ft)2
+ (3 ft)2
= 10.4403 ft
Applying the first theorem of Pappus-Guldinus gives the area:
A = 2 rcL
= 2 (1.5 ft)(10.4403 ft)
= 98.3975 ft2
Calculate the volume of paint required:
Volume of paint = 2(98.3975 ft2
)( )
= 0.656 gal Ans.
Because both the inside and outside surfaces must be painted, the value of
the computed area must be doubled.
1 gal
300 ft2
L

10 ft
y
x
3 ft
3 ft
10 ft
C
x
y
3 ft
3
rc = = 1 ft
2
1 The y axis is the axis of rotation.
The generating area is
a triangle
3 The centroid of a
triangle is located
one-third of the
distance from the base
to the opposite vertes.
4 The area of the triangle is
A = (1/2)(10 ft)(3 ft)
= 15 ft2
5 Applying the second theorem of
Pappus-Guldinus gives the volume:
V = 2 rcA
= 2 ft)( 15 ft2
)
= 94.2 ft3
Ans.
2. Determine the volume of the cone.

4 m
x
y
z
1 m
4 m
x
y
1 m
C
C
3. Determine the area of the
half-torus (half of a doughnut). 1 The axis of revolution is the x axis.
2 The generating curve
is a circle.
3 The distance to the centroid
is rc = 4 m.
4 The length of
the curve is the
circumference
of the circle:
L = 2 (1m)
= 6.2832 m
5 Applying the first theorem of
Pappus-Guldinus gives the area:
A = rcL
= (4 m)(6.2832 m)
= 79.0 m2
Ans.
The angle of revolution is , not 2 , because
the figure is a half -torus.

4 m
x
y
1 m
C
4. Determine the volume of the
half-torus (half of a doughnut).
1 The axis of revolution is the x axis
2 The generating area is the
region bounded by a circle
3 The distance to the centroid is rc = 4m
4 The area bounded by
the circle is
A = (1 m)2
= m2
5 Applying the second theorem of
Pappus-Guldinus gives the volume:
V = rcA
= (4 m)( m2
)
= 39.5 m3
Ans.
The angle of revolution is , not 2 , because
the figure is a half -torus.
4 m
x
y
z
1 m
C

2 m
3 m
x
y
O
O
D
C
B
4 m
2 m
d
3 m
2 m
3 m
x
y
O
B
D
d
5. Determine the area of the frustum of the cone. 1 The y axis is the axis of rotation
2 The generating curve is the
straight line BD. The
horizontal coordinate d of
the lower end of the line
can be found by similar
triangles:
d
2 m
3 m
2 m + 4 m
=
Solving gives
d = 1 m
4 m 4 m

2 m
3 m
x
y
3 m
x
y
O
B
C
D
1 m
rc = (1 m + 3 m)/2
= 2 m
B
C
D
3 m 1 m = 2 m
4 m
O
3 The distance to
the centroid is
4 The length of the generating curve BD is
L = (2 m)2
+ (4 m)2
= 4.4721 m
5 Applying the first theorem of Pappus-Guldinus gives the area:
A = 2 rcL
= 2 (2 m)(4.4721 m)
= 56.2 m2
Ans.
4 m
B
D

6. Determine the volume of the frustum of the cone.
2 m
3 m
x
y
O
4 m
O
D
B
4 m
2 m
d
3 m
1 The y axis is the axis of rotation
trapezoid BDEF. The
horizontal coordinate d of
the lower end of the line
BD can be found by similar
triangles:
d
2 m
3 m
2 m + 4 m
=
Solving gives
d = 1 m
2 m
3 m
x
y
O
B
D
C
d
rC
3 The distance rc to the centroid of the
area can be calculated by dividing the
crosshatched trapezoid into composite
parts and using the formula
rc = (1)
where xel is the centroidal coordinate
of the part with area Ael.
Ael
xelAel
E
F
4 m

However, we can save some work by noting that the
second theorem of Pappus-Guldinus involves the
product of rc and the generating area A:
V = 2 rcA (2)
where V is the volume of the solid of revolution.
Solving Eq. 1 for the product of rc and Ael gives
rc Ael = xelAel
and substituting this result in Eq. 2 gives
V = 2 xelAel (3)
Thus we do not need to calculate rc and A
independently we need only evaluate the first
moment of the area, xelAel.
A = total area = sum of individual elements of area
4

O
4 m
3 m
O
4 m
1 m
1 m
y
x
y
x
O
4 m
2 m
y
x
C
C
1 m
4 m
xel = 1 m + (1/3)(2 m)
= 1.6667 m
xel = 0.5 m
= +
5 To calculate xelAel, divide the trapezoidal region into
the sum of a rectangle and triangle, and set up a table.
6 Substituting the value of xelAel into Eq. 3 gives the volume of the solid:
V = 2 xelAel (Eq. 3 repeated)
= 54.5 m3
Ans.
8.6668
Region Ael (m2
) xel (m) xelAel (m3
)
Rectangle 4 0.5 2
Triangle 4 1.6667 6.6668
xelAel = 8.6668
Ael = (4 m)(1 m)
= 4 m2
Ael = (1/2)(4 m)(2 m)
= 4 m2

y
x
C
rC
R
7. Determine the centroidal coordinate rc of a
semicircular arc of radius R, given that the area of a
sphere of radius R is known to be 4 R2
. 1 If the semicircle is revolved around the y axis,
a sphere of radius R is generated. The first
theorem of Pappus-Guldinus says that the
area of the sphere is given by
A = 2 rcL
Because we already know A (= 4 R2
), we can
solve this equation for rc in terms of R and L:
rc = (1)
The length L in Eq. 1 is the circumference of
the semicircle:
L = R
Substituting this result in Eq. 1 gives
rc = (Eq. 1. repeated)
= Ans.
2 L
A
4 R2
2
4 R2
2 L
R
2R

y
x
C
rC
Radius = R
8. Determine the centroidal coordinate rc of a
semicircular area of radius R, given that the volume of a
sphere is known to be (4/3) R3
.
1 If the semicircular area is revolved around the y axis,
a sphere of radius R is generated. The second
theorem of Pappus-Guldinus says that the volume of
the sphere is given by
V = 2 rcA (1)
Because we already know V (= (4/3) R3
), we can
solve Eq. 1 for rc in terms of A and R:
rc = (2)
V
2 A
(4/3) R3
The area A in Eq. 2 is the area bounded by the
semicircle:
R2
Substituting this result in Eq. 2 gives
rc =
= Ans.
(4/3) R3
2 A
2
A =
R2
2
4R
3
2
(Eq. 2 repeated)

20 m
1 m3 m 2 m
40°
9. A concrete dam is to be constructed in the shape
shown. Determine the volume of concrete that
would be required.
1 m3 m 2 m
y
x
The y axis is the axis of revolution
2 The crosshatched region is the generating area.
1
3.5 m
3.5 m
20 m

A = sum of element areas
The second theorem of Pappus-Guildinus gives the
volume as
V = 2 40°/360°)rcA (1)
where rc is the distance to the centroid of the generating
area, A is the magnitude of the area, and the
factor 40°/360° accounts for the fact that the dam
corresponds to 40° rather than to a complete circle.
Thus we must calculate the product rcA. This product
may be found by dividing the crosshatched area into
composite parts and then using the formula
rc = (2)
where xel is the centroidal coordinate of the part with
area Ael. Solving for the product rcA gives
rcA = xelAel
Thus Eq. 1 can be written as
V = 2 40°/360°)(rcA) (Eq. 1 repeated)
= (2/9) xelAel (3)
and, to calculate V, we have to evaluate the sum xelAel.
3
Ael
xelAel
Generating area divided into a
rectangle and two triangles
1 m3 m 2 m
y
20 m
rc
3.5 m
x

4 Calculate the areas and centroidal
distances, and set up a table.
1 m3 m 2 m
y
x
20 m
rc
3 m
y
O1 m 2 m
y
20 m
23 m
2 m
y
20 m
xel = 23 m + (3 m)/3 = 24 m
xel = 20 m + 2 m + 0.5 m = 22.5 m
xel = 20 m + (2 m)(2/3) = 21.333 m
=
+
+
O
O
O
O
Triangle 1
Triangle 2
Rectangle
3.5 m
3.5 m
3.5 m
3.5 m
Ael = (1/2)(3.0 m) 3.5 m) = 5.25 m2
Ael = (3.5 m) m) = 3.5 m2
Ael = (1/2)(3.5 m) m)
= 3.5 m2
x
x
x

5 Table
6
Region Ael (m2
) xel (m) xelAel (m3
)
Triangle 1 5.25 24 126
Rectangle 3.5 22.500 78.750
Triangle 2 3.5 21.333 74.665
xelAel = 279.415
Substituting the value of xelAel into Eq. 3 gives the
volume of the solid:
V = xelAel (Eq. 3 repeated)
279.415 m3
= 195.1 m3
Ans
2
9

y
x
190 mm
190 mm
260 mm 260 mm
10. The concrete steps shown are in the shape of a quarter
circle. Determine the amount of paint required to paint
the steps, if one liter of paint covers 1.5 m2.
1 The y axis is the axis of revolution.
2 The generating
curve is a series of
four straight
line-segments.
190 mm
190 mm
260 mm
260 mm
y
x
z

L = sum of segment lengths
The first theorem of Pappus-Guildinus gives the area as
A = 2 (90°/360°)rcL
= ( /2)rcL (1)
curve, L is the length of the area, and the factor (90°/360°)
accounts for the fact that the steps are in the shape of a
quarter circle. Thus we must calculate the product rcL.
This product may be found from the formula for the centroid
of a composite curve made of a of collection of line
segments:
rc = (2)
where xel is the centroidal coordinate of the line segment
with length Lel. Solving for the product rcL gives
rcL = xelLel
A = ( /2)(rcL) (Eq. 1 repeated)
= ( /2) xelLel (3)
and, to calculate A, we have to evaluate the sum xelLel.
y
x
190 mm
190 mm
260 mm 260 mm
3
xelLel
Lel

4 Calculate the centroidal coordinates
and lengths of the line segments, and
set up a table.
+
y
x
190 mm
190 mm
260 mm 260 mm
y
x
xel = 260 mm/2
= 130 mm
260 mm
380 mm
y
x
xel = 260 mm
y
x
190 mm
190 mm
260 mm 260 mm
190 mm
y
x
260 mm 260 mm
190 mm
xel = 260 mm + 260 mm
= 520 mm
xel = 260 mm + (260 mm)/2 = 390 mm
+
+
=
Line 1 Line 2
Line 3 Line 4
Lel = 260 mm
Lel = 190 mm
Lel = 190 mm
Lel = 260 mm

6 Substituting the value of xelAelinto Eq. 3 gives the area of the steps
A = xelLel (Eq. 3 repeated)
= 445 164 mm2
283 400 mm2
7 Amount of paint required = 445 164 mm2
[1 m/(1000 mm)]2
[1 liter of paint/1.5 m2
of covered area]
= 0.297 liter Ans.
2
5 Table
Region Lel (mm) xel (mm) xelLel (mm2
)
Line 1 260 130 33 800
Line 2 190 260 49 400
Line 3 260 390 104 100
Line 4 190 520 98 800
xelLel = 283 400

y
x
190 mm
190 mm
260 mm 260 mm
11. The concrete steps shown are in the shape of a quarter circle.
Determine the total number of cubic meters of concrete required to
construct the steps.
2 The generating area is the crosshatched region
shown.
The y axis is the axis of revolution1
190 mm
190 mm
260 mm
260 mm
y
x
z

3
volume as
V = 2(90°/360° rcA
= ( /2)rcA (1)
area, A is the magnitude of the area, and the factor
(90°/360°) accounts for the fact that the steps form a
quarter circle. Thus we must calculate the product rcA.
This product may be found by dividing the crosshatched
area into composite parts and then using the formula
(2)
rcA = xelAel
V = ( /2)(rcA) (Eq. 1 repeated)
= ( /2) xelAel (3)
Ael
xelAel
Generating area divided into two
rectangles
y
x
190 mm
190 mm
260 mm 260 mmO
rc =

4 Calculate the centroidal distances and areas
and set up a table.
5 Substituting the value of xelAelinto Eq. 3 gives the volume of the solid
V = xelAel
= 50 438 270 mm3
= 0.0504 m3
Ans.
32 110 000 mm3
y
x
190 mm
190 mm
260 mm 260 mm
y
x
260 mm
y
x
260 mm 260 mm
O
O O
= +380 mm
2
xel = 260 mm/2 = 130 mm
xel = 260 mm + 260 mm/2 = 390 mm
Rectangle 1
Rectangle 2
190 mm
Region Ael (mm2
) xel (mm) xelAel (mm3
)
Rectangle 1 98 800 130 12 844 000
Rectangle 2 49 400 390 19 266 000
xelAel = 32 110 000
Ael = (190 mm)(260 mm)
= 49 400 mm2
Ael = (380 mm)(260 mm)
= 98 800 mm2

Front view Side view
12. Determine the mass of the steel V-belt pulley
shown. The density of the steel is 7840 kg/m3
.
x
y
1 The x axis is the axis
of revolution:
2 The crosshatched area is
the generating area for
the right half of the
pulley.
100 mm
15 mm
15 mm
70 mm
4 mm 4 mm
12 mm 10 mm 12 mm
(20 mm)/2 = 10 mm
20 mm

3
volume as
V = 2 rcA (1)
area, and A is the magnitude of the area. Thus we must
calculate the product rcA. This product may be found
by dividing the crosshatched area into composite parts
and then using the formula
rc = (2)
rcA = yelAel
V = 2 (rcA) (Eq. 1 repeated)
= 2 yelAel (3)
and, to calculate V, we have to evaluate the sum yelAel.
Ael
yelAel
rectangles, and a triangle
(70 mm 20 mm)/2 = 25 mm
15 mm
x
y
10 mm
4 mm
5 mm 12 mm

Ael = (1/2)(12 mm)(15 mm)
= 90 mm2
Ael = (4 mm)(15 mm)
= 60 mm2
Generating area divided into
two rectangles, and a triangle
10 mm
25 mm
15 mm
x
y
10 mm
4 mm
5 mm 12 mm
12 mmy
4 Calculate the centroidal distances
and areas for the rectangles and
triangle, and set up a table.
x
y
x
4 mm
x
y
5 mm + 12 mm + 4 mm
= 21 mm
25 mm
15 mm
10 mm + 5 mm = 35 mm
15 mm
35 mm
Rectangle 1 Triangle
Rectangle 2
=
+
+
yel = 10 mm + 25/2 mm
= 22.5 mm
yel = 35 mm + 15/3 mm
= 40 mm
yel = 35 mm + 15/2 mm
= 42.5 mm
Ael = (21 mm)(25 mm)
= 525 mm2

Table5
6 Substituting the value of yelAelinto Eq. 3 gives half the volume of the
V-belt pulley
V = yelAel
= 112 861.7 mm3
= 0.000 112 861 7 m3
17 962.5 mm3
Total mass of the V-belt = total volume density
= (2 0.000 112 861 7 m3
) 7840 kg/m3
= 1.770 kg Ans.
7
Region Ael (mm2
) yel (mm) yelAel (mm3
)
Rectangle 1 525 22.5 11 812.5
Triangle 90 40.0 3 600.0
Rectangle 2 60 42.5 2 550.0
yelAel = 17 962.5
Double the half-volume.

y
z
13. Determine the area of the surface of
revolution generated by rotating the curve
y = z4
, 0 z 1 m, about the z axis.
The z axis is the axis of revolution.
The generating curve is
y = z4
, 0 z 1
1
2
1 m
1 m
x
y
1 m
z

3 The first theorem of Pappus-Guildinus gives the area of
the surface of revolution as
A = 2 rcL (1)
curve, and L is the length of the curve. Thus we must
calculate the product rcL. This product may be found
by considering the equation for the centroidal
coordinate:
rc =
L
where dL is an increment of curve length, and yel is the
coordinate of the increment. Solving for the product rcL
gives
rcL = yel dL
A = 2 (rcL)
= 2 yel dL (2)
and, to calculate A, we have to evaluate a single integral,
yeldL.
yel dL
dL
C
dL
y
z
1 m
1 m
y
z
1 m
1 m
rc
yel

dL
dy
dz
5 Evaluating the integral by use of the integral
function key on a calculator gives
A = 3.44 m2
Ans.
To evaluate the integral yel dL in Eq. 2, we use the
equation of the curve,
y = z4
(3)
to express dL as a function of z. Thus
dL = (dy)2
+ (dz)2
= ( )2
+ 1 dz (4)
and differentiating Eq. 3 gives
= 4z3
so Eq. 4 can be written as
dL = (4z3
)2
+ 1 dz (5)
Substitute this expression for dL into Eq. 2
A = 2 yel dL (Eq. 2 repeated)
= 2 z4
(4z3
)2
+ 1 dz
4
dz
dy
dz
dy
0
1
dL
y
z
yel = y = z4
1 m

x
y
1 m
z
y = z4
1 The z axis is the axis of rotation.
area under the y = z4
curve.
14. Determine the volume of the solid of
revolution generated by rotating the curve
y = z4
, 0 z 1 m, about the z axis.
y
z
1 m
1 m

3y
z
rC
volume as
V = 2 rcA (1)
by considering the equation for the centroidal distance:
rc =
A
where dA is an increment of area, and yel is the
coordinate of the centroid of the incremental region.
Solving for the product rca gives
rcA = yel dA
V = 2 (rcA)
= 2 yel dA (2)
and, to calculate V, we have to evaluate a single integral,
yel dA.
yel dA
dA
C
y
z
y = z4
y = z4
(x, y)
yel

y
z
y = z4
(y, z) yel = y/2 = z4
/2
dz
4 To evaluate the integral yel dL in Eq. 2, we use the
equation of the curve,
y = z4
to express dA as a function of z. Thus
dA = y dz
= z4
dz
Substitute this expression for dA into Eq. 2
A = 2 yel dz (Eq. 2 repeated)
= 2 (z4
/2)(z4
) dz
Evaluating the integral by use of the integral function key
on a calculator gives
V = 0.349 m2
Ans.
0
1
1 m

Radius = 20 mm
7 mm
15. A pharmaceutical company plans to put a coating
0.01 mm thick on the outside of the pill shown.
Determine the amount of coating material required.
1 The x axis is the axis of revolution.
The generating curve for half of the
pill surface is a composite curve
consisting of one straight line and a
circular arc. By symmetry, the total
surface area of the pill will be two
times the area generated by the curve
above.
2
x
y
3.5 mm
(1.5 mm)/2 = 0.75 mm
1.5 mm

The first theorem of Pappus-Guildinus gives the area as
A = 2 rcL (1)
curve and L is the length of the curve. Thus we must
calculate the product rcL. This product may be found
by dividing the curve into composite parts and then
using the formula
rc = (2)
length Lel. Solving for the product rcL gives
rcL = yelLel
A = 2 (rcL) (Eq. 1 repeated)
= 2 yelLel (3)
and, to calculate A, we have to evaluate the sum yelLel.
3
Generating curve divided into a
straight line segment and an arc
x
y
3.5 mm
Lel
yelLel
L
0.75 mm

4 Calculate the centroidal distances and
the lengths, and set up a table.
y
=
3.5 mm
20 mm
x
y
yel = 3.5 mm
+
Arc
Straight line
x
y
3.5 mm
x
y
yel = 3.5 mm
Straight line
Radius = 20 mm
0.75 mm
x
5 For the straight line, the length and
coordinate of the centroid are easily
calculated..
Lel = 0.75 mm
0.75 mm0.75 mm

6 To calculate xelLel for the arc,
use the information shown below,
which has been taken from a table
of properties of common
geometric shapes.
x
y
For our particular arc, r = 20 mm and
= (1/2) sin-1
(3.5/20)
= 0.08795 rad
Thus
Length = 2 r
= 2(0.08795) 20 mm
= 3.5180 mm
rarc = (r sin
= (20 mm)(sin 0.08795)/(0.08795)
= 19.9742 mm
yel = rarc sin
19.9742 mm (sin 0.08795)
= 1.7545 mm3.5 mm
20 mm
rarc
yel
Arc
7
2
20 mm
3.5 mm
A
B
C
B C
A
Circular arc segment
r sin
C
Length = 2 rr
x
y
Centroid Location

9 Substituting the value of yelAelinto Eq. 3 gives the area of the solid
A = 2 2 yelLel
= 110.5501 mm2
where a factor of 2 has been inserted to account for the fact that we took advantage
of symmetry to calculate the area of only half of the body.
Amount of coating material required = 110.5501 mm2
0.01 mm
= 1.106 mm3
Ans.
8.7973 mm2
8 Table
Region Lel (m) yel (m) yelLel (m2
)
Line 0.75 3.5 2.6250
Arc 3.5180 1.7545 6.1723
yelLel = 8.7973

10 mm
5 mm
70 mm
60 mm
16. Determine the volume of the funnel.
x
y
The y axis is the axis of revolution1
5 mm
2.5 mm
crosshatched area shown.
5 mm
70 mm
60 mm
10 mm

A
volume as
V = 2 rcA (1)
area and A is the magnitude of the area. Thus we must
by dividing the cross-hatched area into composite parts
rc = (2)
rcA = xelAel
= 2 xelAel (3)
3
Ael
xelAel
rectangles and two triangles
x
y
5 mm
2.5 mm
70 mm
60 mm
10 mm

4 To calculate xelAel, divide the area
into the sum of two rectangles and
two triangles, and set up a table.
=
+ +
Triangle 1
Triangle 2
Rectangle 1
60 mm
x
y
60 mm
x
y
5 mm
10 mm
+
Rectangle 1
O O
xel = 2.5 mm
5 mm
xel = 5 mm + (5 mm)/3
= 6.6667 mm
xel = 2.5 mm + (2.5 mm)/3
= 3.3333 mm
x
y
2.5 mm
x
y
2.5 mm
2.5 mm
70 mm
70 mm
Ael = (5 mm)(60 mm)
= 300 mm2
Ael = (2.5 mm)(70 mm)
= 175 mm2
Ael = (1/2)(5 mm)(60 mm)
= 150 mm2
Ael = (1/2)(2.5 mm)(70 mm)
= 87.5 mm2
xel =1.25 mm
x
y
5 mm
2.5 mm
70 mm
60 mm
10 mm

5
Substituting the value of xelAelinto Eq. 3 gives the volume of the funnel
V = xelAel (Eq. 3 repeated)
= 14 200 mm3
Ans.
2260.4187 mm3
Table
6
Region Ael (mm2
)
Rectangle 1 300 2.5 750
Rectangle 1 175 1.25 218.75
Triangle 1 150 6.6667 1000.0050
Triangle 2 87.5 3.3333 291.6637
xelAel = 2260.4187

0.2 m
Signals parallel
to axis of dish
17. A satellite dish is shaped in the form of a paraboloid of
revolution to take advantage of the geometrical fact that all
signals traveling parallel to the axis of the paraboloid are
reflected through the focus. Determine the amount, in m2
,
of reflecting material required to cover the inside surface of
the dish.
y
x
The x axis is the axis of revolution1
The generating
curve is a parabola.
2
( 0.2, 0)
(0, 0.3)
The general form for a parabola with vertex on the x
axis is
x = ay2
+ b
Evaluating this equation at the points ( 0.2, 0) and
(0, 0.3) gives the equations
0.2 = a(0)2
+ b (1)
and
0 = a(0.3)2
+ b
Solving for a and b and substituting back in Eq. 1
gives
x = 0.2222y2
0.2 (2)
3
0.3 m
0.3 m

x
To evaluate the integral yel dL in Eq. 4, we have to use
the equation of the parabola,
x = 0.2222y2 0.2 (Eq. 2 repeated)
to express dL as a function of y. Thus
dL = (dx)2
+ (dy)2
= 2
+ 1 dy (5)
Differentiating Eq. 2 gives
= 0.4444y,
so Eq. 5 can be written as
dL = (0.4444y)2
+ 1 dy (6)
5
C
( 0.2, 0)
(0, 0.3)
Apply the first theorem of Pappus-Guldinus
to calculate the surface area of the dish:
A = 2 rcL (3)
Thus we must calculate the product of the
length of the parabolic curve and its
centroidal coordinate. The product may be
found by considering the equation for the
centroidal coordinate:
rc =
L
Solving for the product rcL gives
rcL = yel dL
A = 2 (rcL)
= 2 yel dL (4)
4
yel dL
dL
rc
y
dy
dx
dy
dx
dx
dy
dL
dL
yel

Noting that y = yel and also using Eq. 6 to replace dL in
Eq. 4 gives
y
A = 2 yel dL (Eq. 4 repeated)
= 2 y (0.4444y)2
+ 1 dy
Using the integral function on a calculator gives
A = 0.284 m2
Ans
6
x
C
( 0.2, 0)
(0, 0.3)
rc
y
0
0.3
(x, y)
yel

90 mm
80 mm
18. Determine the amount of coffee that
the coffee mug holds when full to the
brim. The radius of the rounded corners
and the rim is 15 mm.
Radius = 15 mm
x
y
Radius = 15 mm
The y axis is the axis of revolution, and the
generating area is the cross-hatched area shown.
1
2 The distance rc to the centroid of the area can
be calculated by dividing the crosshatched
area into composite parts and using the
formula
rc = (1)
where xel is the centroidal coordinate of the
part with area Ael.
Ael
xelAel

However, we can save some work by noting that the second
theorem of Pappus-Guldinus involves the product of rc and the
generating area A:
V = 2 rcA (2)
where V is the volume of solid of revolution.
Solving Eq. 1 for the product gives
rc Ael = xelAel
and substituting this result in Eq. 2 gives
V = 2 xelAel (3)
Thus we do not need to calculate rc and A independently we
need only evaluate the first moment of the area, xelAel.
A
3

4 To calculate xelAel, divide the area into the algebraic
sum of a rectangle, two squares, and two quarter circles,
and set up a table.
Radius = 15 mm
x
y
Radius = 15 mm
x
y
= +
Rectangle x
y
x
y
x
y
x
y
For clarity, the areas near the rim and
rounded corner have been drawn
disproportionately large.
+
90 mm
40 mm
Square 1
Square 2Quarter circle 1 Quarter circle 2
Subtract quarter-
circular area
from square to
form area near
rim.
Subtract square
from quarter-
circular area to
form rounded
corner.
5
6

15 mm
Square 1
Square 2
x
Radius = 15 mm
90 mm
40 mm
Radius = 15 mm
Original area
Ael = (15 mm)2
= 225 mm2
Ael = (40 mm)(90 mm)
= 3600 mm2
15 mm
40 mm
xel = 40 mm (15 mm)/2
= 32.5 mm 15 mm
xel = 40 mm + (15 mm)/2
= 47.5 mm
y
7 Calculate the areas and centroidal coordinates of the
rectangle and the squares.
x
y
Rectangle
40 mm
90 mm
xel = 20 mm
x
y
x
y
40 mm 15 mm

40 mm
8 Calculate the area and centroidal coordinates
of the quarter-circular regions..
x
y
x
y
40 mm
15 mm
Quarter circle 1
Quarter circle 2
9 A table of properties of planar regions
gives the information shown below.
4(15 mm)
y
x
r
C
4r
3 3
4r
3
r2
4
A =
Quarter circular region
Centroid location
(15 mm)2
In our particular problem, r = 15 mm, so the
distance to the centroid is
= = 6.3662 mm4
Also, the area is
Ael = = 176.7146 mm2
4
10
xel = 40 mm + 15 mm 6.3662 mm
= 48.6338 mm
xel = 40 mm 15 mm
+ 6.3662 mm
= 31.3662 mm
15 mm

11
Substituting the value of xelAelinto Eq. 3 gives the volume of the solid
V = xelAel
= 454 000 mm3
Ans.
72 323.563 0 mm3
Table
12
Region Ael (mm2
)
Rectangle 3600 20 72 000
Square 1 225 47.5 10 687.5
Quarter circle 1 176.7146 48.6338 8 594.3025
Square 2 225 32.5 7 312.5
Quarter circle 2 176.7146 31.3662 5 542.8655
xelAel = 72 323.5630

Radius = 20 mm
5 mm
17.5 mm
15 mm
19. Determine the capacity of the small bottle of lotion if the
bottle is filled half way up the neck.
Radius = 20 mm
x
y
The y axis is the axis of revolution.1
15 mm
17.5 mm
2.5 mm
crosshatched area shown.
Half of the
neck is filled:
(5 mm)/2

3
A
volume as
V = 2 rcA (1)
by dividing the cross-hatched area into composite parts
rc = (2)
rcA = xelAel
= 2 xelAel (3)
Ael
xelAel
Generating area divided into a rectangle,
two triangles and a circular sector
Radius = 20 mm
x
y
15 mm
17.5 mm
2.5 mm
O

4
E
O
B
CD
O
Before the areas and centroidal coordinates can be
found, we first must find the distances and angles
shown below.
15 mm 20 mm
17.5 mm 20 mm
EB = (20 mm)2
(15 mm)2
= 13.2288 mm
DC = (20 mm)2
(17.5 mm)2
= 9.6825 mm
EOB = cos-1
= 41.4096° (5)
15
20
DOC = cos-1
= 28.9550° (4)
17.5
20
Radius = 20 mm
x
15 mm
17.5 mm
2.5 mm
E
O
C
D
B

Triangle 1 Triangle 2
Rectangle Circular sector
Radius = 20 mm
x
y
15 mm
17.5 mm
2.5 mm
E
O
C
D
B
=
D
CD
O
O O
C
BE
xel = 9.6825 mm/2
= 4.8412 mm
13.2288 mm
9.6825 mm
Ael = (2.5 mm)(9.6825 mm)
= 24.2062 mm2
Ael = (1/2)(15 mm)(13.2288 mm)
= 99.2160 mm2
Ael = (1/2)(17.5 mm)(9.6825 mm)
= 84.7219 mm2
C
B
5 The areas and centroidal coordinates of
the rectangle and triangles can now be
calculated.
x
y
17.5 mm
x
y
15 mm
x
y
2.5 mm
x
y
+
+ +
xel = 9.6825 mm/3
= 3.2275 mm
xel = (13.2288 mm)/3
= 4.4096 mm
9.6825 mm

x
y
= (180° DOC EOB)/2
by Eq. 4 by Eq. 5
= (180° 28.9550° 41.4096°)/2
= 54.8177°
= + EOB 90°
= 54.8177° + 41.4096° 90°
= 6.2273°
rc =
=
= 11.3903 mm
xel = rc cos
= (11.3903 mm) cos 6.2273°
= 11.3231 mm
xel
y
xO
Ael = r2
= ( 54.8177°/180°) 20 mm)2
= 382.6997 mm2
r
C
2r sin
y
x
6 To calculate the area and centroidal
coordinate of the circular sector, we can use
the information shown below, which has been
taken from a table of properties of planar
regions. Note that in the formula equals
half the angle of the arc.
A = r2
Circular sector region
C
BE
D
2r sin
3
2(20 mm) sin 54.8177°
3(54.8177° /180°)
rc
DOC = 28.9550°
EOB = 41.4096°
7
8
r = 20 mmCentroid Location

9
Substituting the value of xelAel into Eq. 3 gives the capacity of the bottle:
V = xelAel
= 32 400 mm3
Ans.
5161.4768 mm3
Table
10
Region Ael (mm2
)
Triangle 1 84.7219 3.2275 273.4399
Triangle 2 99.2160 4.4096 437.5029
Rectangle 24.2062 4.8412 117.1871
Circular sector 382.6997 11.3231 4333.3470
xelAel = 5161.4768

Theorem pappus (1)

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