PROBLEM 5.34
Determine by direct integration the centroid of the area shown. Express
your answer in terms of a and h.
SOLUTION
We have
and
dA (h
h 1
xEL
yEL
Then A
and
h
x
a
y
y )dx
x
dx
a
x
1
(h y )
2
h
x
1
2
a
a
dA
xEL dA
yEL dA
0
a
0
h 1
x
dx
a
x
x h 1
dx
a
a
0
h2
2
h
x
1
2
a
a
0
1
h x
x2
2a
x2
h
2
h 1
x
dx
a
x2
dx
a2
h2
x
2
a
1
ah
2
0
a
x3
3a
1 2
a h
6
0
x3
3a 2
a
0
1 2
ah
3
xA
xEL dA: x
1
ah
2
1 2
a h
6
x
2
a
3
yA
yEL dA: y
1
ah
2
1 2
ah
3
y
2
h
3
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PROBLEM 5.40
Determine by direct integration the centroid of the area shown. Express
your answer in terms of a and b.
SOLUTION
At
b
( x a)2
a2
y
Then
Now
dA
ydx
Then A
dA
and
and
xEL dA
xEL
x
yEL
y
2
0,
y
b
b
k (0 a) 2
or k
b
a2
b
( x a)2
2
2a
b
( x a)2 dx
2
a
a
0
a
0
x
b
( x a) 2 dx
a2
a
0
b
3a 2
b
( x a) 2 dx
a2
b x4
a2 4
yEL dA
x
2 3
ax
3
b
a2
a2 2
x
2
x a
a
0
3
x3
a
0
1
ab
3
2ax 2
a 2 x dx
1 2
a b
12
b
b
( x a) 2 2 ( x a) 2 dx
2
a
2a
b2 1
( x a )5
4
5
2a
a
0
1 2
ab
10
Hence
xA
xEL dA: x
1
ab
3
1 2
a b
12
yA
yEL dA: y
1
ab
3
1 2
ab
10
x
y
1
a
4
3
b
10
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PROBLEM 5.44
Determine by direct integration the centroid of the area shown.
Express your answer in terms of a and b.
SOLUTION
For y1 at x
Then y1
and for 0
For a
x a,
a
dA
0
b
( x 2b) b 2
a
yEL
1
y1
2
b 2
x
a2
and dA
y EL
1
y2
2
b
2
2
x
a
xEL dA
2b 2
x dx
a2
a
2b x3
a2 3
and
y2
x
a
x
x 2a,
Then A
2b
a2
2b 2
x
a2
By observation,
Now xEL
ka 2 , or k
2b, 2b
y
a,
0
0
x
a
a
2
2
b
a
2a
2b 2
x dx
a2
2b x 4
a2 4
a
b x2
0
1 2
a b b
2
7 2
a b
6
(2a) 2
2a
2
0
2a
a
y2 dx
b 2
x
dx
a
7
ab
6
x b 2
x3
3a
and dA
2b 2
x dx
a2
x
dx
a
b 2
x
a
y1 dx
x
dx
a
2a
0
(a)2
1
(2a 2 ) (a )3
3a
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PROBLEM 5.44 (Continued)
yEL dA
a
0
b 2 2b 2
x
x dx
a2
a2
2b2 x5
a4 5
a
0
b2
2
2a
0
a
2
3
b
2
2
x
a
3
x
a
b 2
x
dx
a
2a
a
17 2
ab
30
Hence,
xA
xEL dA: x
7
ab
6
7 2
a b
6
yA
yEL dA: y
7
ab
6
17 2
ab
30
x
y
a
17
b
35
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