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10.1 Moments of Inertia by Integration
10.1 Moments of Inertia by Integration Example 1, page 1 of 2
1. Determine the moment of inertia of the rectangle
about the x axis, which passes through the centroid C.
C
b
h/2
x
y
h/2
10.1 Moments of Inertia by Integration Example 1, page 2 of 2
x
yWe want to evaluate
Ix = y2 dA
where the differential element dA is located a
distance y from the x axis (y must have the same
value throughout dA).
yh/2
h/2
Ix = y2 dA
= y2 (b dy)
by3
3
bh3 Ans.
12
=
=
Limits of integration
dA = area of rectangle
= b dy
1
2
3
4h 2
-h 2
dy
h 2
-h 2
h 2
-h 2
C
b
10.1 Moments of Inertia by Integration Example 2, page 1 of 2
h
b
y
x
2. Determine the moment of inertia of the rectangle
about its base, which coincides with the x axis.
10.1 Moments of Inertia by Integration Example 2, page 2 of 2
b
y
x
We want to evaluate
Ix = y2 dA
where the differential element of area dA is located a distance y from the
x-axis (y must have the same value throughout the element dA).
hy
2 dA = area of rectangle
= b dy
3
=
0
h
Ix = y2 dA
= y2 (b dy)
by3
3
= Ans.
Limits of integration4
1
h
0
dy
bh3
3
10.1 Moments of Inertia by Integration Example 3, page 1 of 3
b
3. Determine the moment of inertia of the
right triangle about the x axis.
x
y
h
10.1 Moments of Inertia by Integration Example 3, page 2 of 3
x
y
b
x
y
(x,y)
We want to evaluate
Ix = y2 dA
where y has the same value throughout
differential element dA.
Equation of line
y = slope x + intercept
( ) x + h (1) =
1
dA = area of rectangle
= x dy
h
Since we will integrate with respect to y,
we must replace x by a function of y.
Solve for x to get
x = ( ) y + b
so,
dA = x dy
= ( y + b) dy
4
5
3
2
dy
b
h
bh
hb
10.1 Moments of Inertia by Integration Example 3, page 3 of 3
Ix = y2 dA
= y2 ( y + b) dy
= b ( + y2) dy
= b[ + ]
= bh3[ + ]
= Ans.
6
Limits of integration are from
bottom of triangle to top.7
h
0
bh
y3
h
y4
4h
y3
3
h
0
311
4
bh3
12
0
h
10.1 Moments of Inertia by Integration Example 4, page 1 of 3
x
yx2 y2
a2 b2 = 1+
4. Determine the moments of inertia of the area
bounded by an ellipse about the x and y axes.
b
b
a a
10.1 Moments of Inertia by Integration Example 4, page 2 of 3
x
y
dy
x
y
We want to evaluate
Ix = y2 dA
where y has the same value throughout the
differential element dA.
x = half the length of the
differential element,
so 2x = entire length
1
(x, y) dA = area of rectangle
= 2x dy
Since we will integrate with
respect to y, we must replace x
by a function of y
Since x locates a point to the right of the y axis, choose
the plus sign:
x = +a 1 (y/b)2 (1)
+ 1=a2 b2
x2 y2
Solve for x to get
x = a 1 (y/b)2
2
4
3
6
5
10.1 Moments of Inertia by Integration Example 4, page 3 of 3
Evaluate the integral either with a
calculator that does symbolic
integration or use a table of integrals.
To calculate Iy, use symmetry in the following way: in all the
above equations, replace x's by y's, original y's by x's, a's by b's
and original b's by a's. Then the result would be
Iy = Ans.
Limits of integration from
bottom to top of ellipse
8
9
11
10
b
-b
4ab3
a3 b4
7 Using Eq. 1 in the expression for dA gives:
dA = 2x dy
= 2a 1 (y/b)2 dy
Ix = y2 dA
= y2(2a 1 (y/b)2 ) dy
= Ans.
10.1 Moments of Inertia by Integration Example 5, page 1 of 4
2 ft
5. Determine the moments of inertia of the
crosshatched area about the x and y axes.
y = 4 x2
x
y
4 ft
10.1 Moments of Inertia by Integration Example 5, page 2 of 4
2 ft
We want to evaluate
I x = y2 dA
where y has the same value throughout the differential element dA.
dA = area of rectangle
= x dy
Since we will integrate with respect to y,
we must replace x by a function of y.
Solving for x to get
x = 4 y
Since x locates a point to the right of the y axis, choose the plus sign:
x = + 4 y (1)
1
2
3
4
5
y = 4 x2
(x,y)
y
dy
x
y
4 ft
10.1 Moments of Inertia by Integration Example 5, page 3 of 4
Using Eq. 1 in the expression for dA gives
dA = x dy
= 4 y dy
6
Ix = y2 dA
= y2 4 y dy
= 19.50 ft4 Ans.
Use the integral function
on your calculator
7
8
9
y = 4 at the top of
the crosshatched area
4
0
10.1 Moments of Inertia by Integration Example 5, page 4 of 4
2 ft
Next we want to evaluate
Iy = x2 dA
where x has the same value throughout the differential element dA. Thus, we
choose a vertical rectangular strip.
dA = area of rectangle
= y dx
= (4 x2) dx
Iy = x2 dA = x2(4 x2 ) dx = 4.27 ft4 Ans.
dx
x
10
11
120
2
yy = 4 x2
x
y
4 ft
10.1 Moments of Inertia by Integration Example 6, page 1 of 3
6. Determine the moment of inertia of the
crosshatched area about the y axis.
x = 2y6 50y5 y3 + 100
x
y
1.156 m
100 m
Scales on
the x and y axes are
not the same.
10.1 Moments of Inertia by Integration Example 6, page 2 of 3
100 m
1 We want to evaluate
Iy = x2 dA
where x has the same value throughout the differential
element dA. Thus, we choose a vertical rectangular strip.
dA = area of rectangle
= y dx
But this approach won't work because we
can't express y as a function of x.
2
3
y
x = 2y6 50y5 y3 + 100
x
y
10.1 Moments of Inertia by Integration Example 6, page 3 of 3
4 An alternative approach is to use a horizontal
rectangular strip and employ the equation for the
moment of inertia of a rectangle about its base (BB) :
(1)
B
Bh
b
100 m
1.156 m
yApplying Eq.1 to the differential element
gives the differential moment of inertia.
dIy
(dy)x3
= 3
5
Replacing x by the function of y gives
dIy = x3dy
= (2y6 50y5 y3 + 100)3 dy
Iy = dIy = (2y6 50y5 y3 + 100)3 dy
= 2.72 105 m4 Ans.
Use the integral function on a calculator to evaluate
this integral.
6
7
8
x = 2y6 50y5 y3 + 100
x
x
dy
bh3
3I BB =
=3bh3
31
31
31
0
1.156
10.1 Moments of Inertia by Integration Example 7, page 1 of 3
y = 3x x2
y = 9 x2
7. Determine the moment of inertia of the
crosshatched area about the x axis.
9 m
3 m
x
y
10.1 Moments of Inertia by Integration Example 7, page 2 of 3
x
y
We want to evaluate
Ix = y2 dA
where y has the same value throughout the differential
element dA. Thus it appears that we should use a
horizontal rectangular strip.
But using a horizontal strip is awkward we would
have to use three different expressions for dA,
depending on the position of the strip.
2
1
A better approach is to use a vertical strip and then
apply the parallel-axis theorem to the strips.
Parallel-axis theorem
for a general region
Ix = Ic + Ad2d
Centroid
Area A
x
y
For a rectangle in particular,
Ix = bh3 + (b h)d2
= ( h3 + hd2 )b
y
x
y
x
C
d
C
h
b
h
b dx
d
The moment of inertia of a strip of width
b dx is then
dIx = ( h3 + hd2 ) dx (1)
3
121
112
121
C
10.1 Moments of Inertia by Integration Example 7, page 3 of 3
x
y
(xel,yel)
(x,y1)
(x,y2)
The y-coordinate of the element
centroid is the average of y1 and y2
d = yel = ( y1 + y2 )
Eq. 1 becomes
dIx = ( h3 + hd2 ) dx
= { ( y2 y1 )3 + ( y2 y1 ) [ ( y1 + y2 )]
2} dx
Since the point (x, y2) lies on the curve y = 9 x2, we
can substitute
y2 = 9 x2
in Eq. 2. Similarly, since (x, y1) satisfies y = 3x x2, we
can substitute
y1 = 3x x2
in Eq. 2. Thus Eq. 2 becomes
5 6
h = y2 y1
4
yel
centroid
y = 3x x2
y = 9 x2
dIx = { ( y2 y1 )3 + ( y2 y1 ) [ ( y2 + y1 )]
2} dx
= { [( 9 x2 ) ( 3x x2 )]3 + [( 9 x2 )
( 3x x2 )][ ( 9 x2 ) + ( 3x x2 )]2} dx
= { ( 9 3x )3 + ( 9 3x )[ + x x2 ]2} dxIx = dIx
= { ( 9 3x )3 + (9 3x) ( + x x2 )2 } dx
= 328 m4 Ans.
Enter this expression directly into the
integral function of a calculator.
7
8
x ranges
from 0 to 3.
21 12
1
112 2
1
121 1
2112
29 3
292 2
30
3121
112
122
1
(2)
10.1 Moments of Inertia by Integration Example 8, page 1 of 4
8. Determine the moment of inertia of the
crosshatched area about the y axis.
xy = 1
y = 2x4 m
1 m1 m1/2 m
4 m 1/4 m
y
x
10.1 Moments of Inertia by Integration Example 8, page 2 of 4
We want to evaluate
Iy = x2 dA
where x has the same value throughout the differential
element dA. Thus it appears that we should use
vertical differential strips.
But using a vertical strip is awkwardwe would have to use three different expressions for dA, depending on the location of the strip.
2
1
x
y
10.1 Moments of Inertia by Integration Example 8, page 3 of 4
3 A better approach is to use a horizontal strip and then
apply the parallel-axis theorem to the strip.
x
y
x
y
y
x
area A
centroid
C
Parallel-axis theorem for a general region
Iy = Ic + Ad2
For a rectangle in particular,
Iy = bh3 + (b h)d2
= ( h3 + hd2 )b
The moment of inertia of a strip of width b dy is then
dIy = ( h3 + hd2 ) dy (1)
C
C
d
d
d
Area = b h
b
h
b dy
h
12
112
112
1
10.1 Moments of Inertia by Integration Example 8, page 4 of 4
| xel | = | average of x1 and x2 | = | ( x1 + x2 )|d =
h = x2 x1
y
x
xy = 1
y = 2x
2
y
2
y
112 4
y
2y
1
2
y2
yy1
y14
1y
412y
112 2
y1y
y
21y
1y2
1y
2y1
121
y1
12
112
1y
21
121
121
21
y
2
8
7
6
5
Top of region at y = 4
Bottom of
region at y = 1= 2.81 m4 Ans.
Enter this expression directly into the
integral function of a calculator
Iy = dIxy
= [ ( + )3 + ( + ) ( )2 ] dy
dIy = { ( x2 x1 )3 + ( x2 x1 )[ ( x2 + x1 )]
2 } dy
= { ( ( ))3 + ( ( ))[ ( + ( ) )]2 } dy
= { ( + )3 + ( + )[ ]2 } dy
Since the point (x1,y) lies on the line y = 2x,
we can solve for x and substitute
x1 =
in Eq. 2. Similarly, since (x2,y) lies on the
curve xy = 1, we can substitute
x2 =
in Eq. 2. Thus Eq. 2 becomes
4
Centroid of
strip
From the figure, we see that Eq. 1 can be written as
dIy = { h3 + hd2} dy (Eq. 1 repeated)
= { ( x2 x1 )3 + ( x2 x1 )[ ( x2 + x1 )]
2 } dy (2)
(x2,y)(x1,y)
(xel,yel)
C
10.1 Moments of Inertia by Integration Example 9, page 1 of 3
9. Determine the moment of inertia of the
crosshatched area about the x axis.
y
x
y = 10e x2
x2ey = 10
1 m
1.5 m
10.1 Moments of Inertia by Integration Example 9, page 2 of 3
1
2
3 Ix for region D
No integration needed.
Use the formula for the moment of inertia of a
rectangle about a centroidal axis.
4
b
h/2
h/2
So
=
=
C
IxD
12
bh3
Ic = (1)
12
33.1915 m4 (2)
x
y
3.6788 m
3.6788 m
1 m
(1 m)(2 3.6788 m)3
y(1) = 10e = 3.6788 m(1)2
1 m
We want to evaluate
Ix = y2dA
Because of the shape of the region, the integral has to
be evaluated over two sub-regions, D and E.
Region E
Region D
10.1 Moments of Inertia by Integration Example 9, page 3 of 3
5 Ix for region E. Because the region is
symmetrical about the x axis, we can save
work by using vertical rectangular strips and
by applying Eq. 1 to these strips:
Ic = bh3
dIxE = (dx)(2y)3
= (dx)[2( )]3
= dx
Thus, with the aid of the integral function on a
calculator, we have
IxE = dIx
= e dx
= 4.7985 m4 (3)
Adding the results for regions D and E gives
Ix = IxD + IxE
= 33.1915 + 4.7985
= 38.0 m4 Ans.
by Eq. 2 by Eq. 3
6121
112
112
20003
1
1.5
32000
or,
x
y
dx
1 m
x
1.5 m
10e x2
x2ey = 10
3x2e
3x2
10.1 Moments of Inertia by Integration Example 10, page 1 of 3
1 in.
2 in.
4 in.
x
y
y = x2
y = 3x
10. Determine the moment of inertia of the
crosshatched area about the y axis.
3 in.
10.1 Moments of Inertia by Integration Example 10, page 2 of 3
1
2
3 Integrate over region B.
y
x
x
1 in.
(x,y2)
(x,y1)
y = 3x
y = x2
dA = area of rectangle
= ( y2 y1 ) dx
= ( 3x x2 ) dx
4
5 IyB
= x2 dA
= x2(3x x2) dx
= 0.5500 in4 (2)
1
0
dx
y
x
y2 y1
We want to evaluate
Iy = x2dA (1)
where x has the same value throughout the differential
element dA.
Given the shape of the area, we have to evaluate the
integral in Eq.1 over two sub-regions, B and C.
Region C
Region B
10.1 Moments of Inertia by Integration Example 10, page 3 of 3
x
y
6 Integrate over region C.
4 in.
x
y2 y1
dx
2 in.
(x,y1)
y = x2
(x,y2)
Write the equation of the line: =
Solving gives
y = x + 2
dA = Area of rectangle
= ( y2 y1 ) dx
= [(x + 2) x2 ] dx
IyC
= x2 dA
= x2( x + 2 x2 ) dx
= 2.2167 in4 (3)
Adding the results for regions B and C gives
Iy = IyB
+ IyC
= 0.5500 + 2.2167 = 2.77 in4 Ans.
by Eq. 2 by Eq. 3
7
8
9
10
2 1
x 1 y 3
4 3
1
2
1 in.
3 in.
10.1 Moments of Inertia by Integration Example 11, page 1 of 2
Area of quarter circle
4a2 4a
3
4a4
9
34a
a
11. Given that the centroid C of the area bounded by a
quarter-circle lies a distance 4a/(3 above the base of the
quarter-circle, determine the moment of inertia of the cross
hatched area about an axis xc through the centroid.
y yc
xc
x
x2 + y2 = a2
C
We want to evaluate
Ixc = yc2 dA
but because the equation of the circle is given in
terms of x and y instead of xc and yc, it is easier to
evaluate
Ix y2 dA
and then use the parallel-axis theorem:
Ix = Ixc + Ad2
= Ixc + ( )2
Solving gives
Ixc = Ix
Thus now we need to calculate Ix.
1
10.1 Moments of Inertia by Integration Example 11, page 2 of 2
x2 + y2 = a2
x
y2
y
x
(x,y)
dA = Area of rectangle
= x dy
Solve x2 + y2 = a2
to get
x = a2 y2
Substitute into the equation for dA
dA = x dy
= a2 y2 dy
Integrate
Ix = y2 dA
= y2 a2 y2 dy
= (2)
a
0
Use the result given by Eq. 2 in Eq. 1:
IxC = Ix (Eq. 1 repeated)
( ) a4 Ans.
Evaluate the integral with a
calculator that does symbolic
integration or use a table.
3
4
5
6
7
dy
16
4a4
9
94a4 4
9a4
16
a
a4
16