SME
Calculus II · Problem type
Volume from the centroid
Use this for “The region under y = x² from 0 to 3 turns about the x-axis. Use Pappus’s theorem to find the volume.”
Diagram
The region under y = x² on [0, 3], turned about the x-axis.
Its area is A = 9, with its centroid ȳ = 2.7 from the axis.
The centroid travels a circle 2πȳ = 2π × 2.7 = 16.96.
Pappus: V = 2πȳA = 2π × 2.7 × 9 = 152.7.
This is an example. Type your own values to start a new problem.
y
=
c
x
,
0
≤
x
≤
b
V
=
2πȳA
=
A
Area of the region
worked out (not typed)
ȳ
Centroid’s height
worked out (not typed)
L
Path of the centroid (2πȳ)
worked out (not typed)
Formulas
A
=
c
n
+ 1
×
b
n
+ 1
9
=
1
2 + 1
×
3
2 + 1
ȳ
=
(
n
+
1)
×
c
×
b
n
2 × (2 ×
n
+ 1)
2.7
=
(2
+
1)
×
1
×
3
2
2 × (2 × 2 + 1)
L
=
2
×
π
×
ȳ
16.9646 = 2 × π × 2.7
V
=
L
×
A
152.6814 = 16.9646 × 9
Assumptions
•
Pappus: a flat region turned a full circle about a line in its plane, and not crossing it, sweeps V = 2πȳA.
•
Here the region under y = c·xⁿ from 0 to b turns about the x-axis, so ȳ is its centroid’s height, Mₓ ÷ A.
•
The centroid travels a circle 2πȳ long. The disk method’s ∫πf(x)² dx gives the same volume, which makes a check.
Step-by-step
WE KNOW
c = 1, n = 2, b = 3
FIND
area of the region (A), centroid’s height (ȳ), path of the centroid (2πȳ) (L), volume of the solid (V)
Step 1 · Find area of the region (A)
Use A = c ÷ (n + 1) × bⁿ⁺¹
A
=
b
∫
0
cxⁿ dx
:
raise
the
power
by
one
and
divide
by
the
new
power.
A
=
1
2 + 1
×
3
2 + 1
3
∫
0
x
2
dx
=
[
x
3
3
]
from
0
to
3
=
9
A
=
9
Step 2 · Find centroid’s height (ȳ)
Use ȳ = (n + 1) × c × bⁿ ÷ (2 × (2 × n + 1))
Mₓ
=
1
2
∫
c
2
x
2
ⁿ
dx
=
c
2
b
2
ⁿ⁺¹
÷
(2(
2
n
+
1)).
Divided
by
A
=
cbⁿ⁺¹
÷
(
n
+
1),
one
c
and
bⁿ⁺¹
cancel.
ȳ
=
(2
+
1)
×
1
×
3
2
2 × (2 × 2 + 1)
ȳ
=
3
×
1
×
9
2 × 5
ȳ
=
27
10
ȳ
=
2.7
Step 3 · Find path of the centroid (2πȳ): L
Use L = 2 × π × ȳ
The
centroid
turns
on
a
circle
of
radius
ȳ
about
the
x-axis,
2πȳ
around.
L
=
2
×
π
×
2.7
L
=
5.4
×
π
L
=
16.9646
Step 4 · Find volume of the solid (V)
Use V = L × A
Pappus:
the
region
sweeps
its
area
along
the
path
of
its
centroid,
so
V
=
2πȳ
×
A
.
V
=
16.9646
×
9
Check
by
disks:
∫
from
0
to
3
of
π
×
x
4
dx
=
[π
×
x
5
5
]
from
0
to
3
=
152.6814
V
=
152.6814
Check
Put the numbers back in. Both sides should match.
9
=
1
2 + 1
×
3
2 + 1
✓
2.7
=
(2
+
1)
×
1
×
3
2
2 × (2 × 2 + 1)
✓
16.9646 = 2 × π × 2.7 ✓
152.6814 = 16.9646 × 9 ✓
← Previous lesson
Force on an upright rectangular gate
Next lesson →
Numerical integration (trapezoid and Simpson's rules)
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.