9.1 Centroids by Integration

9.1 Centroids by Integration Procedures and Strategies, page 1 of 2

x

y

dx

y = f (x)

(x, y)

(xel, yel)

xel = x

Procedures and Strategies for Solving Problems

Involving Calculating Centroids by Integration

1. Determine the coordinates of the centroid by

evaluating integrals such as

xc =

For a planar area, the differential area dA is usually a

rectangular strip of finite length and differential width dx

(for a vertical strip) or dy (for a horizontal strip). Use a

vertical strip if the curve bounding the planar region is

given as a function of x, y = f(x). Use a horizontal strip

if the bounding curve is given as a function of y, x = g(y).

The integrand xel is the x coordinate of the centroid of the

strip. It must be expressed as a function of x for a

vertical strip and as a function of y for a horizontal strip.

xel dA

dA

dy

x

y

x = g(y)

(xel, yel)

(x, y)

xel = x/2

= g(y)/2

9.1 Centroids by Integration Procedures and Strategies, page 2 of 2

dL

dx

dy

x

y2. For a line (a wire), the area element dA is replaced by

dL = (dx)2 + (dy)2)

= 1+ (dy/dx)2 dx

if the line is given as a function of x: y = f(x). Use

dL = (dx/dy)2 + 1 dy

if the line is given as function of y: x = g(y).

3. For volumes with some degree of symmetry (for example, a solid

of revolution), dA can be replaced by a circular disk of finite radius

and differential thickness.

4. Using the integral function on a scientific graphing calculator

simplifies the work and helps avoid errors.

z = f(x)

x

y

z

Radius = x

dy

9.1 Centroids by Integration Problem Statement for Example 1

x

1. Locate the centroid of the plane area shown. Use a

differential element of thickness dx.

y

y = 3x2

2 ft

12 ft

9.1 Centroids by Integration Problem Statement for Example 2

x

y

y = a sin( )2b

x

a

b

2. Locate the centroid of the plane area shown, if a = 3 m

and b = 1 m. Use a differential element of thickness dy.

9.1 Centroids by Integration Problem Statement for Example 3

y

x1 in.

3. Locate the centroid of the plane area shown.

1 in

13 in.

y = 4x5 3x2 + 12x + 1

9.1 Centroids by Integration Problem Statement for Example 4

xy = 1

x

y

4. Locate the centroid of the plane area shown.

0.5 m

2 m

0.5 m

2 m

9.1 Centroids by Integration Problem Statement for Example 5

5. Locate the centroid of the plane area shown.

x

y =

y

y = x2 +

x(13 x)

6

14 11x3

6 m

2 m

1 m

4 m

9.1 Centroids by Integration Problem Statement for Example 6

x = 3yx = 4 y2

y

x

6. Locate the centroid of the plane area shown.

1 m

3 m 1 m

9.1 Centroids by Integration Problem Statement for Example 7

y

x

7. Locate the centroid of the plane area shown. Use a

differential element of thickness dx.

y = h b

x

h

b

9.1 Centroids by Integration Problem Statement for Example 8

x = a[1 ( )2]

y

x

y

b

b

a

8. Locate the centroid of the plane area shown. Use a

differential element of thickness dy.

9.1 Centroids by Integration Problem Statement for Example 9

9. A sign is made of 0.5 in. thick steel plate in the shape shown.

Determine the reactions at supports B and C.

x = 50 + (10) sin

Specific weight = 490

B

C

y24

lbft3

y

x

50 in.

72 in.

9.1 Centroids by Integration Problem Statement for Example 10

y = 2x2

x

y

10. Locate the centroid of the wire shown.

3 m

18 m

9.1 Centroids by Integration Problem Statement for Example 11

x = 300[1 ( )4]

y

x

11. Locate the centroid of the wire shown.

y200

300 mm

200 mm

9.1 Centroids by Integration Problem Statement for Example 12

12. The rod is bent into the shape of a circular arc.

Determine the reactions at the support A.

3 ft

20°

0.2 lb/ft

A

9.1 Centroids by Integration Problem Statement for Example 13

y

x

625 ft

299 ft 299 ft

13. a) Locate the centroid of the Gateway Arch in St.

Louis, Missouri, USA. b) During the pre-dawn hours of

September 14, 1992, John C. Vincent of New Orleans,

Louisiana, USA, climbed up the outside of the Arch to the

top by using suction cups and then parachuted to the

ground. Estimate the length of his climb.

Approximate equation of centerline:

y = 639.9 ft (68.78 ft) cosh[(0.01003 ft-1)x]

9.1 Centroids by Integration Problem Statement for Example 14

Ox

y

3 m

14. Locate the centroid of the cone shown.

z

Radius = 2 m

9.1 Centroids by Integration Problem Statement for Example 15

y

x

One-eighth of a

sphere of radius "a"

a

15. Locate the centroid of the volume shown.

z

9.1 Centroids by Integration Problem Statement for Example 16

(This curve is rotated about the

x-axis to generate the solid.)

16. Determine the x coordinate of the centroid of the solid

shown. The solid consists of the portion of the solid of

revolution bounded by the xz and yz planes.

z

y

x = a[1 ( z b

)2]

x

a

b

9.1 Centroids by Integration Problem Statement for Example 17

x

y

z

a a

b

b

17. Locate the centroid of the pyramid shown.

h