Applications of Integration
Integrals compute physical totals. Learn area between curves, volumes of revolution (disc and shell), arc length and surface area, work done by variable forces, mass, centroids and moments, average and RMS values, and charge and energy accumulation.
Integration turns a density or rate into a total: mass from density, work from force, energy from power, centroid positions from area moments — the numbers engineers use to size parts and systems.
An analogy
Integration is slicing an object into very thin pieces, computing each piece easily, and adding them all up.
Areas and volumes
Subtract bottom from top curve; find intersections for limits. Rotate a curve about an axis for volumes: discs (perpendicular slices) or shells (cylindrical layers).
Physical quantities
Work = ∫F dx (springs, pumping liquids), hydrostatic force = ∫ρgh w(h) dh, mass = ∫ρ dV, centroids and second moments for beams (I = ∫y² dA).
Averages and RMS
The average of a sinusoid over a full cycle is 0; over a half cycle 2/π × peak. RMS = peak/√2 — the heating-equivalent value used for AC power.
Choosing the slice
Pick slices where the integrand is simple and limits are easy. Check units (m × m² = m³) and sanity (volume positive, centroid within the shape).
Formulas
- Area between curves:
A = ∫ₐᵇ (f_top − f_bottom) dx - Disc method:
V = π ∫ₐᵇ [f(x)]² dx - Arc length:
L = ∫ₐᵇ √(1 + (y′)²) dx - Work:
W = ∫ F(x) dx (spring: ½kx²) - Centroid:
x̄ = ∫ x dA / ∫ dA - Average & RMS:
f_avg = (1/(b − a))∫f dx; f_rms = √((1/T)∫f² dt)
Worked example: Volume of revolution
Given: y = √x from 0 to 4 rotated about the x-axis
Find: Volume.
- V = π∫₀⁴ x dx = π × 8
Answer: 8π ≈ 25.1 units³
Worked example: Spring work
Given: k = 200 N/m, stretch 0 → 0.1 m
Find: Work.
- W = ∫₀^0.1 200x dx = 100 × 0.01
Answer: 1 J
Common mistakes
- Wrong: Using (f − g)² instead of f² − g² for washers. — Right: Washer: π∫(R² − r²) dx.
- Wrong: Forgetting ρg in hydrostatic problems. — Right: Pressure = ρgh at depth h.
Applications
- Structural design: Centroids and second moments of area.
- Tanks and dams: Volumes and hydrostatic forces.
- Electrical power: RMS and average values.
Key takeaways
- Slice, compute, sum.
- Area, volume, arc length formulas.
- Work, mass, centroid via integrals.
- Average and RMS values.
Going further
- Pappus's theorems: Volumes via centroid path.
- Moments of inertia: ∫r² dm for rotating bodies.
- Multiple integrals: Mass of 2D/3D bodies (M3).