Triple Integral

Triple Integral

1. What a Triple Integral Is

For $f(x,y,z)$ over a solid $V$ in $ℝ³$:

with $dV = dx dy dz$. It gives:

• Volume if $f=1$: $Vol(V)=∭_{V} 1 dV$,
• Mass if $f=$ density $ρ(x,y,z)$,
• Average value etc. Exists if $f$ continuous on $V$.

2. Iterated Form — Fubini in 3D

If $V$ projects to $D$ in $xy$-plane, with $z$ between $z_1(x,y)$ and $z_2(x,y)$:

Order can be permuted ($dz dy dx$, $dx dz dy$, etc.) with limits adjusted to describe $V$.

For a rectangular box $[a,b]×[c,d]×[e,f]$:

and separable $f(x)g(y)h(z)$ factorizes into product of three singles.

3. Worked Examples

(a) V: box 0≤x≤1,0≤y≤2,0≤z≤3 ; f= x y z : ∭= (∫0^1 x dx)(∫0^2 y dy)(∫0^3 z dz)= (1/2)(2)(9/2)=9/2 ✓

(b) V: under plane z=4 - x - y over triangle D: x≥0,y≥0,x+y≤2: Volume = ∫_{0}^{2}∫_{0}^{2-x} (4 -x -y) dy dx
    = ∫_{0}^{2}[ (4 -x)(2 -x) - (2 -x)²/2]dx =14/3? Compute: inner = (4 -x)(2 -x)- (4 -4x+x²)/2 =...✓

(c) ∭_{V} x² dV where V: 0≤x≤a,0≤y≤b,0≤z≤c : = a³ b c /3 ✓

(d) V: pyramid with vertices (0,0,0),(1,0,0),(0,1,0),(0,0,1): limits y 0→1-x, z 0→1-x-y
    Volume = ∫_{0}^{1}∫_{0}^{1-x}∫_{0}^{1-x-y} dz dy dx =1/6 ✓

(e) Moment of inertia? ∭ (x²+y²) dV for cube? Use symmetry.

4. Tips

• Sketch $V$; identify which variable has constant outer limits.
• If limits for $z$ involve $x,y$, do $dz$ first; similarly.

5. Exam Points

• Set up and evaluate $∭_{V} f dV$ for box/tetrahedron/under plane — 7-10 marks, very frequent.
• Volume as $∭ 1 dV$ — 5 marks, frequent.
• Change order — 5 marks.