Applications of Integrals — Volume, Physics & More

The definite integral measures accumulation. Once you have that idea, the applications follow naturally: area is accumulated height, volume is accumulated cross-sectional area, arc length is accumulated distance, work is accumulated force. The specific formulas differ, but the underlying logic is always the same Riemann sum taken to its limit.

This page covers the main geometric and physical applications. Each one sets up as an integral — the hard part is always the setup, not the evaluation.

Volumes of Revolution — Disk Method

• Rotate y=f(x) around the x-axis. Each cross-section is a disk with radius f(x) and area π[f(x)]². Volume: V = π∫ₐᵇ [f(x)]² dx.
• Example: Volume of sphere radius r = π∫₋ᵣʳ (r²−x²)dx = (4/3)πr³.

Arc Length

The length of a curve y = f(x) from a to b is found by summing infinitesimal arc elements. Each element has length ds = √(dx² + dy²) = √(1 + (dy/dx)²) dx. Therefore: L = ∫ₐᵇ √(1 + [f'(x)]²) dx. This formula requires f' to be continuous on [a,b]. Computing arc length often produces integrals with no closed form — they must be evaluated numerically.

Work Done by a Variable Force

For constant force F over displacement d, Work = F·d. When force varies with position: W = ∫ₐᵇ F(x)dx. Applications: stretching a spring (Hooke's Law: F = kx, so W = ∫₀^x₀ kx dx = kx₀²/2), pumping water from a tank (force = weight of water at each height × distance to pump), lifting a chain (force decreases as more chain is raised).

Fluid Pressure and Force

The pressure at depth h in a fluid of density ρ is P = ρgh. The force on a horizontal plate at depth h with area A is F = P·A = ρghA. For a vertical plate, pressure varies with depth, requiring integration: F = ∫ ρg·h(y)·w(y)dy where w(y) is the width of the plate at depth y. This arises in dam design, submarine hull engineering, and fluid storage tank calculations.

Probability and the Normal Distribution

For a continuous random variable X with probability density function f(x), P(a ≤ X ≤ b) = ∫ₐᵇ f(x)dx. The normal distribution with mean μ and standard deviation σ has density f(x) = (1/σ√(2π))·e^(−(x−μ)²/(2σ²)). Computing probabilities from the normal distribution — such as P(μ−σ ≤ X ≤ μ+σ) ≈ 0.6827 (the 68% rule) — requires evaluating an integral with no closed form, done numerically using the error function erf(x). Every statistical test, confidence interval, and hypothesis test in data science uses these integrals.