1. Problem 9.185 y y2 = mx Determine by direct integration
y1 = kx2 y
Determine by direct integration
the moments of inertia of the
shaded area with respect to
the x and y axes.

2. Solving Problems on Your Own
Determine by direct integration
the moments of inertia of the
shaded area with respect to
the x and y axes.
Problem b
y2 = mx a x
y1 = kx2 y
1. Calculate the moments of inertia Ix and Iy. These moments of
inertia are defined by:
Ix = y2 dA and Iy = x2 dA
Where dA is a differential element of area dx dy.
1a. To compute Ix choose dA to be a thin strip parallel to the x
axis. All the the points of the strip are at the same distance y from
the x axis. The moment of inertia dIx of the strip is given by y2 dA.

3. Solving Problems on Your Own
Determine by direct integration
the moments of inertia of the
shaded area with respect to
the x and y axes.
Problem b
y2 = mx a x
y1 = kx2 y
1b. To compute Iy choose dA to be a thin strip parallel to the y
axis. All the the points of the strip are at the same distance x from
the y axis. The moment of inertia dIy of the strip is given by x2 dA.
1c. Integrate dIx and dIy over the whole area.

4. m = y1 = k x2: b = k a2 k = y1 = k x2 then: x1 = y1 1/2
Problem Solution b
y2 = mx a x
y1 = kx2 y
Determine m and k:
At x = a, y2 = b: b = m a
m =
y1 = k x2: b = k a2
k = b a b a2
Express x in terms of y1 and y2:
y1 = k x then: x1 = y1
y2 = mx then x2 = y2 1
k1/2 m
1/2

5. dA = ( y2 - y1 ) dx = ( m x _ k x2 ) dx
Problem Solution y a
To compute Iy choose dA to
be a thin strip parallel to the y
axis. b y2
dA = ( y2 - y1 ) dx
= ( m x _ k x2 ) dx y1 x x dx a
Iy = x2 dA = x2 ( m x _ k x2 ) dx
= ( m x3 _ k x4 ) dx = [ m x4 _ k x5 ]
= m a4 _ k a5 = b a3 a 1 4 1 5 a 1 4 1 5 1 20
Iy = b a3/ 20
Substituting k = , m = b a2 a

6. dA = ( x1 - x2 ) dy = ( y _ y ) dy 1/2 Ix = y2 dA = y2 ( y _ y ) dy
Problem Solution y a
To compute Ix choose dA to
be a thin strip parallel to the x
axis. x2 dy b
dA = ( x1 - x2 ) dy
= ( y _ y ) dy 1
k1/2 m
1/2 y x x1
Ix = y2 dA = y2 ( y _ y ) dy
= ( y _ y3 ) dy = [ y _ y4 ]
= b _ b4 = a b3 1
k1/2
1/2 m
5/2 b 2 7
7/2 4 28
Ix = a b3/ 28
Substituting k = , m = b a2 a