Integration
Indefinite and definite integration, areas under curves
Integration is the reverse of differentiation. If dy/dx = xⁿ, then y = xⁿ⁺¹/(n+1) + c (where c is the constant of integration, n ≠ −1).
Indefinite integrals: ∫3x² dx = x³ + c. Always add + c.
Definite integrals: ∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a). No + c needed.
Area under a curve: The definite integral gives the area between the curve, the x-axis, and the limits. If the curve is below the x-axis, the integral is negative — take the absolute value.
Area between two curves: ∫ₐᵇ [f(x) − g(x)] dx where f(x) is above g(x).
Integration to find equations: Given dy/dx = 6x² − 4x and a point (1, 5):
y = 2x³ − 2x² + c. Substituting (1,5): 5 = 2 − 2 + c → c = 5. So y = 2x³ − 2x² + 5.
Volumes of revolution: V = π∫ₐᵇ y² dx (rotation about x-axis).
Key Points to Remember
- 1∫xⁿ dx = xⁿ⁺¹/(n+1) + c
- 2Definite integral: F(b) − F(a), no constant
- 3Area under curve = definite integral
- 4Below x-axis gives negative area — take absolute value
Pakistan Example
Water Flow Through Tarbela Dam — Integration in Engineering
Engineers at Tarbela Dam calculate total water flow by integrating the flow rate over time. If flow rate is r(t) = 500 + 100sin(t) cubic metres per second, total water in 6 hours = ∫₀²¹⁶⁰⁰ r(t) dt. This is real A Level integration applied to Pakistan's largest dam.