Rucete ✏ AP Calculus In a Nutshell
8. Further Applications of Integration
This chapter explores how integrals are used to solve applied problems involving motion, rates of change, and accumulated quantities. The concept of Riemann Sums is extended to real-world problems, including economics, biology, and environmental science.
A. Motion Along a Straight Line
• If position is s = F(t), then:
– Velocity: v(t) = F′(t) – Acceleration: a(t) = v′(t) = F″(t)
• Distance traveled from t = a to t = b is:
∫ab |v(t)| dt
• Displacement (net position change): ∫ab v(t) dt
Example: v(t) = t³ + 3t² → Since v(t) ≥ 0 on [1, 4], distance = ∫14 v(t) dt
Example: v(t) = 6t² – 18t + 12 – Break into intervals around turning points (t = 1, 2) – Use ∫|v(t)| dt for total distance – Use ∫v(t) dt for displacement
Example: Acceleration Given
• If a(t) = sin t and v(0) = 0 → integrate twice to get s(t)
• Distance = ∫0π v(t) dt, where v(t) = ∫a(t) dt
B. Motion Along a Plane Curve (BC)
• For motion with parametric equations x = x(t), y = y(t):
– Position vector: R(t) = ⟨x(t), y(t)⟩
– Velocity vector: v(t) = ⟨dx/dt, dy/dt⟩ – Speed = |v(t)| = √[(dx/dt)² + (dy/dt)²]
• Distance traveled from t₁ to t₂:
∫t₁t₂ √[(dx/dt)² + (dy/dt)²] dt
Example: Projectile motion – Start at origin, initial angle α, velocity v₀ – Parametric equations: x(t) = v₀ cos α · t y(t) = v₀ sin α · t – (1/2)gt²
Example: Given acceleration and initial conditions → integrate to get velocity and position → convert parametric to Cartesian if needed
C. Riemann Sums in Context
• Approximate total quantity by dividing into n intervals of width Δx → Total ≈ Σ f(xi) Δx → As n → ∞, becomes ∫ f(x) dx
Example: Water draining from a pipe with area A and velocity v(t) → Volume = A ∫0T v(t) dt
Example: Traffic density f(x) (cars per mile) → Total cars on [0, 10] = ∫010 f(x) dx
D. Net Change and Accumulated Change
• If F′(x) = f(x), then net change in F over [a, b]: ∫ab f(x) dx = F(b) – F(a)
• This represents: – Net position (from velocity) – Net profit (from marginal profit) – Net population change (from growth rate)
E. Applications of Accumulated Change
Example: Bacterial Growth • Growth rate: G(t) bacteria/hr • ∫ab G(t) dt gives total growth
Example: Spread of Infection • Rate: I(t) new cases/day • Accumulated cases: ∫0t I(x) dx
Example: Marginal Cost • C′(x) = cost of producing 1 more item • ∫ab C′(x) dx = total additional cost from a to b items
Example: Decay of Chemical • Rate: R(t) mg/hour (negative value) • Remaining amount = Initial – ∫0t |R(t)| dt
Example: Pollution Flow • Rate of pollutant entering river = r(t) g/min • Accumulated pollutant over [a, b]: ∫ab r(t) dt
In a Nutshell
Integrals model cumulative change across diverse fields. From motion to economics, they calculate total distance, cost, growth, or pollution. Real-world applications often convert discrete sums to continuous integrals, using context-based Riemann sums or direct integration of rate functions.