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 ft3 ft

10 ftC

= 1.5 ft3 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

y3 ft

3rc = = 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 mC

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

D3 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

xO

4 m

2 m

y

x

CC

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 (m

3)

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

xC

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

A4 R2

2

4 R2

2 LR

2R


y

xC

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

2A =

R2

24R3

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 (m

3)

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

x260 mm

y

x

260 mm 260 mmO

O O

= +380 mm

2

xel = 260 mm/2 = 130 mm

xel = 260 mm + 260 mm/2 = 390 mm

Rectangle 1Rectangle 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 mm12 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


dLdy

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

dzdy

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

zrC


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

11 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 mm3.5 mm

A

BC

B C

A

Circular arc segment

r sin

C

Length = 2 rr

x

yCentroid 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 mm70 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 mm5 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 mm70 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 mmx

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


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

dydx

dydx

dx

dydL

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 mm215 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 mmx

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 mm15 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

rC

4r3

3

4r3

r2

4A =

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


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

CD

B


Triangle 1 Triangle 2

Rectangle Circular sector

Radius = 20 mm

x

y

15 mm

17.5 mm

2.5 mm

E

O

CD

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


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

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Theorem pappus (1)