Evaluation of Triple integral by changing to Spherical Polar co-ordinates

Evaluation of Triple integral by changing to Spherical Polar co-ordinates

1. Why Spherical

When $V$ is a ball, hemisphere, spherical shell, cone, or integrand involves $x²+y²+z² = r²$, spherical polar $(r,θ,φ)$ simplifies. Definitions (mathematical vs physics swapped — use the convention your notes use; here $θ$ = azimuth in $xy$, $φ$ = colatitude from $z$):

with $r = √(x²+y²+z²)$, $x²+y²+z² = r²$, and Jacobian (volume element):

If your text swaps $θ↔φ$, the $sin$ stays with the angle from $z$; the $r²$ factor remains.

2. Limits

• Full ball $x²+y²+z² ≤ a²$: $r 0→a, φ 0→π, θ 0→2π$.
• Ball octant $x≥0,y≥0,z≥0$: $θ 0→π/2, φ 0→π/2, r 0→a$.
• Hemisphere $z≥0$: $φ 0→π/2$.
• Between spheres $a≤r≤b$: $r a→b$.
• Cone $z = √(x²+y²)$ ⇒ $φ = π/4$ (since $tan φ = √(x²+y²)/z =1$).

3. Worked Examples

(a) Volume of sphere radius a: V= ∫_{0}^{2π}∫_{0}^{π}∫_{0}^{a} r² sinφ dr dφ dθ = 4π a³/3 ✓

(b) ∭_{x²+y²+z²≤a²} (x²+y²+z²) dV = ∫_{0}^{2π}∫_{0}^{π}∫_{0}^{a} r²· r² sinφ dr dφ dθ = 4π·a⁵/5? Actually ∫ r⁴ dr = a⁵/5, angular integral 4π ⇒ 4π a⁵/5 ✓

(c) ∭_{hemisphere z≥0} z dV : = ∫_{0}^{2π}∫_{0}^{π/2}∫_{0}^{a} (r cosφ)·r² sinφ dr dφ dθ = 2π·(a⁴/4)·(1/2)·? Compute: ∫ cosφ sinφ dφ 0→π/2 =1/2, ∫ r³ dr =a⁴/4 ⇒ = π a⁴/4 ✓

(d) Mass of ball with density ρ = k r (proportional to distance from origin): Mass= ∭ k r dV = ∫∫∫ k r·r² sinφ =4π k a⁴/4? Actually ∫ k r³ dr =k a⁴/4 ×4π ⇒ π k a⁴ ✓

(e) I = ∫_{0}^{a}∫_{0}^{√(a²-x²)}∫_{0}^{√(a²-x²-y²)} (x²+y²+z²)^{3/2} dz dy dx (octant): convert ⇒ ∫_{0}^{π/2}∫_{0}^{π/2}∫_{0}^{a} r³·r² sinφ dr dφ dθ = (π/2)(1)·a⁶/6? Actually r³·r²= r⁵? Wait (r²)^{3/2}=r³, times Jacobian r² sinφ ⇒ r⁵ sinφ ⇒ ∫ r⁵= a⁶/6 × (π/2) ✓

4. Tips

• Do not forget $r² sinφ$ — most errors drop it.
• Keep $θ, φ$ order consistent with $dV$ you write.
• For $∭_{octant}$, $θ,φ$ each $0→π/2$.

5. Exam Points

• Convert $∭_{V} f(x,y,z) dV$ over ball/cone to $∭ f(r) r² sinφ dr dφ dθ$ — 7 marks, very frequent.
• Evaluate $∭ (x²+y²+z²)$, $∭ z$ over sphere/hemisphere — 7 marks, frequent.
• Volume via spherical — 5 marks, frequent.