SME
Calculus III (Multivariable) · Problem type
Volume inside a sphere and a cone
Use this for “Find the volume of the solid inside the sphere ρ = 2 and the cone φ = π/3 (60°).”
Diagram
Region: 0 ≤ r ≤ 2, 0° ≤ θ ≤ 60°; dA = r dr dθ; area ½(β − α)(r₂² − r₁²) = 2.09
This is an example. Type your own values to start a new problem.
V
=
2π
3
a
3
(1
−
cos
α
°)
=
The triple integral of ρ² sin φ splits in three: θ gives 2π, ρ gives a³ ÷ 3 and φ gives 1 − cos α.
Vball
Volume of the whole ball
worked out (not typed)
s
Share of the ball
worked out (not typed)
Formulas
Vball
=
4π
×
a
3
3
32π/3
=
4π
×
2
3
3
V
=
(2π
×
a
3
3
)
×
(1
−
cos(
α
°))
8π/3
=
(2π
×
2
3
3
)
×
(1
−
cos(60°))
s
=
V
Vball
0.25 = 8π/3 ÷ (32π/3)
Assumptions
•
In spherical coordinates dV = ρ² sin φ dρ dφ dθ; the factor ρ² sin φ is the Jacobian.
•
The solid is 0 ≤ ρ ≤ a, 0 ≤ φ ≤ α and 0 ≤ θ ≤ 2π, a cone with its tip at the origin and a round cap.
•
The cone angle α is typed in degrees (π/3 is 60°). α = 90° gives the upper half of the ball, α = 180° all of it.
Step-by-step
WE KNOW
a = 2, α = 60°
FIND
volume of the whole ball (Vball), volume inside both (V), share of the ball (s)
Step 1 · Find volume of the whole ball (Vball)
Use Vball = 4π × a³ ÷ 3
The whole ball, for comparison: the same integral with φ running all the way to 180°.
Vball
=
4π
×
2
3
3
Vball = 32π/3 (≈ 33.51)
Step 2 · Find volume inside both (V)
Use V = (2π × a³ ÷ 3) × (1 − cos(α°))
The
limits
are
constants,
so
the
triple
integral
of
ρ
2
sin
φ
splits
into
three
single
integrals
multiplied
together.
V
=
(2π
×
2
3
3
)
×
(1
−
cos(60°))
2
∫
0
ρ
2
dρ
=
[
ρ
3
3
]
from
0
to
2
=
8
3
∫
from
0
to
α
of
sin
φ
dφ
=
1
−
cos(60°)
=
1
2
∫ from 0 to 2π of dθ = 2π
V
=
8
3
×
1
2
×
2π
V
=
8π/3
(≈
8.38)
Step 3 · Find share of the ball (s)
Use s = V ÷ Vball
Divide
by
the
whole
ball.
The
a
3
cancels,
so
the
share
is
1 − cos
α
2
for
a
sphere
of
any
size.
s
=
8π/3
÷
(32π/3)
s
=
0.25
→
The
solid
holds
25%
of
the
ball
Check
Put the numbers back in. Both sides should match.
32π/3
=
4π
×
2
3
3
✓
8π/3
=
(2π
×
2
3
3
)
×
(1
−
cos(60°))
✓
0.25 = 8π/3 ÷ (32π/3) ✓
← Previous lesson
Cylindrical and spherical coordinates, change of variables
Next lesson →
Area under a linear change of variables
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.