SME
Calculus II
Moments, centroids and hydrostatic force
Use this for “Find the centroid of the region under y = x² from x = 0 to x = 3.”
Chart
Shaded: ∫ from 0 to 3 of f(x) dx = 9
This is an example. Type your own values to start a new problem.
y
=
c
x
,
0
≤
x
≤
b
(x̄,
ȳ)
=
(
x̄
,
ȳ
)
A
Area of the region
worked out (not typed)
Mᵧ
Moment about the y-axis
worked out (not typed)
Mₓ
Moment about the x-axis
worked out (not typed)
Formulas
A
=
c
n
+ 1
×
b
n
+ 1
9
=
1
2 + 1
×
3
2 + 1
Mᵧ
=
c
n
+ 2
×
b
n
+ 2
20.25
=
1
2 + 2
×
3
2 + 2
Mₓ
=
c
2
2 × (2 ×
n
+ 1)
×
b
2 ×
n
+ 1
24.3
=
1
2
2 × (2 × 2 + 1)
×
3
2 × 2 + 1
x̄
=
Mᵧ
A
2.25
=
20.25
9
ȳ
=
Mₓ
A
2.7
=
24.3
9
Assumptions
•
The region lies under y = c·xⁿ and over the x-axis from x = 0 to x = b. The plate is uniform, so its density cancels.
•
A strip at x, dx wide, has area f(x) dx and its own centroid at height f(x) ÷ 2, so Mᵧ = ∫x·f(x) dx and Mₓ = ½∫f(x)² dx.
•
The centroid (x̄, ȳ) = (Mᵧ ÷ A, Mₓ ÷ A) is the balance point: the plate rests level on a pin there.
Step-by-step
WE KNOW
c = 1, n = 2, b = 3
FIND
area of the region (A), moment about the y-axis (Mᵧ), moment about the x-axis (Mₓ), centroid x (x̄), centroid y (ȳ)
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 moment about the y-axis (Mᵧ)
Use Mᵧ = c ÷ (n + 2) × bⁿ⁺²
Each
strip’s
area
f
(x)
dx
sits
x
from
the
y-axis,
so
Mᵧ
=
∫x·cxⁿ
dx:
the
power
goes
up
by
two.
Mᵧ
=
1
2 + 2
×
3
2 + 2
3
∫
0
x
3
dx
=
[
x
4
4
]
from
0
to
3
=
20.25
Mᵧ = 20.25
Step 3 · Find moment about the x-axis (Mₓ)
Use Mₓ = c² ÷ (2 × (2 × n + 1)) × b2 × n + 1
Each
strip’s
centroid
is
f
(x)
2
above
the
x-axis,
so
Mₓ
=
∫(
f
(x)
2
)·
f
(x)
dx
=
1
2
∫
c
2
x
2
ⁿ
dx.
Mₓ
=
1
2
2 × (2 × 2 + 1)
×
3
2 × 2 + 1
∫
from
0
to
3
of
x
4
2
dx
=
[
x
5
10
]
from
0
to
3
=
24.3
Mₓ = 24.3
Step 4 · Find centroid x (x̄)
Use x̄ = Mᵧ ÷ A
The centroid’s x is the moment about the y-axis shared over the whole area.
x̄
=
20.25
9
x̄
=
2.25
Step 5 · Find centroid y (ȳ)
Use ȳ = Mₓ ÷ A
The centroid’s y is the moment about the x-axis shared over the whole area.
ȳ
=
24.3
9
ȳ
=
2.7
Check
Put the numbers back in. Both sides should match.
9
=
1
2 + 1
×
3
2 + 1
✓
20.25
=
1
2 + 2
×
3
2 + 2
✓
24.3
=
1
2
2 × (2 × 2 + 1)
×
3
2 × 2 + 1
✓
2.25
=
20.25
9
✓
2.7
=
24.3
9
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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