PROBLEM 5.35
Determine by direct integration the centroid of the area shown.
Express your answer in terms of a and h.
SOLUTION
At
(, ),ah
2
1
:yhka
or
2
h
ka
PROBLEM 5.35 (Continued)
22
24
24
0
22
35
4
0
1
2
11
235
a
a
hh
x
xdx
aa
ha
xx
a









PROBLEM 5.36
Determine by direct integration the centroid of the area shown.
Express your answer in terms of a and h.
SOLUTION
For the element (el) shown at
2, xayh
or

2
2hka
2
= 4
h
ka
el
xx
PROBLEM 5.37
Determine by direct integration the centroid of the area shown.
SOLUTION
First note that symmetry implies
0y
PROBLEM 5.38
Determine by direct integration the centroid of the area shown.
SOLUTION
First note that symmetry implies
0x
For the element (EL) shown
PROBLEM 5.39
Determine by direct integration the centroid of the area shown.
SOLUTION
For the element (EL) shown,
22
b
yax
a

and
()
dA b y dx

PROBLEM 5.39 (Continued)
22 22
0
223
2
22
0
0
2
() 3
22
a
EL
a
a
bb
ydA a a x a a x dx
aa
bbx
xdx
aa

 








PROBLEM 5.40
Determine by direct integration the centroid of the area shown. Express
your answer in terms of a and b.
SOLUTION
At
2
2
0,
(0 ) or
xyb
b
bk a k a

 
Then
2
2
()
b
yxa
a

Now
EL
xx
PROBLEM 5.41
Determine by direct integration the centroid of the area shown. Express
your answer in terms of a and b.
SOLUTION
222
11 1 1 2
333
22 2 2 3
3
2
21 2
but
but
()
b
ykx bka y x
a
b
ykx bkay x
a
bx
dA y y dx x dx
a
a



 



PROBLEM 5.41 (Continued)
33
22
22
0
2
a
EL
bxbx
ydA x x dx
aa
aa






PROBLEM 5.42
Determine by direct integration the centroid of the area shown
.
SOLUTION
We have
2
2
11
22
EL
EL
xx
axx
yy LL

 



PROBLEM 5.43
Determine by direct integration the centroid of the area shown. Express
your answer in terms of a and b.
SOLUTION
For y
2
at
,xa
2
2
,,or
a
yb akb k b
 
Then
1/ 2
2
b
yx
a
PROBLEM 5.43 (Continued)
and
1/ 2 1/ 2
/2
0/2
1
2
aa
EL a
xxx
x
dA x b dx x b dx
a
aa
 

 
 
 
 
/2 5/2 3 4
5/2
0/2
5/2 5/2
5/2
32
32
22
5534
2()
52 2
11
() ()
3242
a
a
a
bxxx
xb a
aa
ba a
a
a
aa
ba a
a





 

 


 
 





 
  


 
 




PROBLEM 5.44
Determine by direct integration the centroid of the area shown.
Express your answer in terms of a and b.
SOLUTION
For y
1
at
,xa
2
2
2
2, 2 , or
b
yb bka k
a
 
Then
2
12
2
b
yx
a
By observation,
2
(2) 2
bx
yxbb
aa

  


PROBLEM 5.44 (Continued)
2
22
22
00
222
2
aa
EL
bb bx x
ydA x xdx b dx
aa
aa





 
PROBLEM 5.45
A homogeneous wire is bent into the shape shown. Determine by direct
integration the x coordinate of its centroid.
SOLUTION
First note that because the wire is homogeneous, its center of gravity coincides with the centroid of the
corresponding line.
Now
322
cos and
EL
x a dL dx dy

where
32
32
cos : 3 cos sin
sin : 3 sin cos
xa dx a d
ya dy a d




PROBLEM 5.45 (Continued)
2/3 2/3
2/3 2/3 3/2
1or ( )
xy ya x
aa
 
 
 
 
Then 2/3 2/3 1/2 1/3
()()
dy ax x
dx
 
PROBLEM 5.46
A homogeneous wire is bent into the shape shown. Determine by direct
integration the x coordinate of its centroid.
SOLUTION
First note that because the wire is homogeneous, its center of gravity coincides with the centroid of the
corresponding line.
Now
cos and
EL
xr dLrd


PROBLEM 5.47*
A homogeneous wire is bent into the shape shown. Determine by direct
integration the x coordinate of its centroid. Express your answer in terms of a.
SOLUTION
First note that because the wire is homogeneous, its center of gravity will coincide with the centroid of the
corresponding line.
We have at
,xa
3/2
1
,,orya aka k a
 
PROBLEM 5.47* (Continued)
Use integration by parts with
3/2
49
2(4 9 )
27
u x dv a x dx
du dx v a x

