1. Problem 5.157 y
y = kx1/3
Locate the centroid of the volume obtained by rotating the shaded area about the x axis. a x h

2. Solving Problems on Your Own
y = kx1/3
Solving Problems on Your Own
Locate the centroid of the volume obtained by rotating the shaded area about the x axis. a
The procedure for locating the
centroids of volumes by direct
integration can be simplified: x h
1. When possible, use symmetry to help locate the centroid.
2. If possible, identify an element of volume dV which produces
a single or double integral, which are easier to compute.
3. After setting up an expression for dV, integrate and determine
the centroid.

3. Problem Solution y
Use symmetry to help locate the centroid. Symmetry implies x dx
y = 0 z = 0 z x r
Identify an element of volume dV which produces a single or double integral.
y = kx1/3
Choose as the element of volume a disk or radius r and thickness dx. Then
dV = p r2 dx xel = x

4. dV = p r2 dx xel = x r = kx 1/3 dV = p k2 x2/3dx a = kh1/3 k = a/h1/3
Problem Solution y x
Identify an element of volume dV which produces a single or double integral. dx
dV = p r2 dx xel = x z x r
Now
r = kx 1/3
so that
y = kx1/3
dV = p k2 x2/3dx
At x = h, y = a :
a = kh1/3 or
k = a/h1/3 a2
h2/3
Then
dV = p x2/3dx

5. ò ò ò [ ] dV = p x2/3dx V = p x2/3dx = p x5/3 = p a2h
Problem Solution y
Integrate and determine the centroid. x dx
dV = p x2/3dx a2
h2/3 ò h a2
h2/3
V = p x2/3dx z x r a2
h2/3
[ ] 3 5 h
= p x5/3
y = kx1/3 3 5
= p a2h a2
h2/3 a2
h2/3 ò ò h 3 8
Also
xel dV = x (p x2/3 dx) = p [ x8/3 ]
= p a2h2 3 8

6. ò ò V = p a2h xel dV = p a2h2 xV = xdV: x ( p a2h) = p a2h2 x = h
Problem Solution y
Integrate and determine the centroid. x dx 3 5
V = p a2h ò 3 8
xel dV = p a2h2 z x r
y = kx1/3 ò 3 5 3 8
Now
xV = xdV:
x ( p a2h) = p a2h2
x = h 5 8
y = 0 z = 0