Chapter 9 Homework Take the negative that y b when root so

CHAPTER 9

PROBLEM 9.1

Determine by direct integration the moment of inertia of the shaded area

with respect to the y axis.

SOLUTION

PROBLEM 9.2

Determine by direct integration the moment of inertia of the shaded area

with respect to the y axis.

SOLUTION

1/3

ykx

For

:

x

a

1/3

bka

PROBLEM 9.3

Determine by direct integration the moment of inertia of the shaded area

with respect to the y axis.

SOLUTION

By observation h

yx

b



PROBLEM 9.4

Determine by direct integration the moment of inertia of the shaded area

with respect to the y axis.

SOLUTION

At

3

0, : xybbka 

PROBLEM 9.5

Determine by direct integration the moment of inertia of the shaded area

with respect to the x axis.

SOLUTION

At



2

0, : 0xybbka 

PROBLEM 9.6

Determine by direct integration the moment of inertia of the shaded area

with respect to the x axis.

SOLUTION

1/3

b

yx

a



PROBLEM 9.7

Determine by direct integration the moment of inertia of the shaded area

with respect to the x axis.

SOLUTION

By observation h

yx

b



PROBLEM 9.8

Determine by direct integration the moment of inertia of the shaded area

with respect to the x axis.

SOLUTION

PROBLEM 9.9

Determine by direct integration the moment of inertia of the

shaded area with respect to the x axis.

SOLUTION

At

3, :xayb

3

(3 )bkaa

or

3

8

b

ka



PROBLEM 9.10

Determine by direct integration the moment of inertia of the

shaded area with respect to the x axis.

SOLUTION

PROBLEM 9.11

Determine by direct integration the moment of inertia of the shaded area

with respect to the x axis.

SOLUTION

PROBLEM 9.12

Determine by direct integration the moment of inertia of the

shaded area with respect to the y axis.

SOLUTION

At

3, :xayb

3

(3 )bkaa

or

3

8

b

ka



PROBLEM 9.13

Determine by direct integration the moment of inertia of the

shaded area with respect to the y axis.

SOLUTION

At

0, :xyb

(1 0)bc

or cb

,0:xa y

1/2

0(1 )cka

1/2

1

or ka



PROBLEM 9.14

Determine by direct integration the moment of inertia of the shaded area

with respect to the y axis.

SOLUTION

At

,:xa yb

/aa

bke

or

b

ke



PROBLEM 9.15

Determine the moment of inertia and the radius of gyration of the

shaded area shown with respect to the x axis.

SOLUTION

21/2

11 2 2

ykx y kx

For

12

0andx yyb

21/2

12

21/2

bka bka

bb

kk

aa



PROBLEM 9.16

Determine by direct integration the moment of inertia of the shaded area

with respect to the y axis.

SOLUTION

At 2, :xayh



2

2hka

or

2

4

h

ka



PROBLEM 9.17

Determine the moment of inertia and the radius of gyration of the

shaded area shown with respect to the y axis.

SOLUTION

See figure of solution on Problem 9.15.

22

21

1()

3y

Aab dIxdAxyydx



PROBLEM 9.18

Determine the moment of inertia and the radius of gyration of the shaded

area shown with respect to the y axis.

SOLUTION

See figure of solution on Problem 9.16

2222

22

7

12

44

y

hh

y x A ah dA ydx dI x dA x x dx

aa









PROBLEM 9.19

Determine the moment of inertia and the radius of gyration of the

shaded area shown with respect to the x axis.

SOLUTION

1

:y

At

2, 0:xay

0sin(2)cka

2or

2

ak k a





At ,:xa yh sin ( )

2

hc a

a



 or ch