definite integrals area

Definite integrals represent the exact accumulated value of a function over a specified interval [a, b]. The Fundamental Theorem of Calculus connects differentiation and integration: ∫ₐᵇ f(x)dx = [F(x)]ₐᵇ = F(b) - F(a), where F(x) is the antiderivative of f(x).

Key Definition: The definite integral ∫ₐᵇ f(x)dx equals the signed area between the curve y = f(x) and the x-axis from x = a to x = b. Areas above the x-axis are positive; areas below are negative.

Essential Properties:

• ∫ₐᵇ f(x)dx = -∫ᵇₐ f(x)dx (reversing limits changes sign)
• ∫ₐᵇ f(x)dx + ∫ᵇᶜ f(x)dx = ∫ₐᶜ f(x)dx (additivity)
• ∫ₐᵇ kf(x)dx = k∫ₐᵇ f(x)dx (constant multiples)
• ∫ₐᵃ f(x)dx = 0 (same limits)

Area Calculations: For area between curves, use A = ∫ₐᵇ |f(x) - g(x)|dx where f(x) ≥ g(x). When finding total area (not signed area), split the integral at x-intercepts and sum absolute values.

Mnemonic: LIFT - Limits matter, Integrate the function, Find F(b) - F(a), Take absolute values for area.

Cambridge Standard: Always show limits of integration clearly and evaluate using square bracket notation [F(x)]ₐᵇ before substituting values. This demonstrates understanding of the definite integral process, essential for method marks.

Think of definite integrals as accumulation machines. Imagine driving a car where your speedometer shows v(t) km/h at time t. The definite integral ∫₀⁵ v(t)dt calculates your total distance travelled in 5 hours—it accumulates all the infinitesimal distances covered at each instant.

Real-World Application 1: Physics & Displacement If velocity v(t) = 3t² - 12t + 9 (m/s), the displacement from t = 0 to t = 4 seconds is ∫₀⁴ (3t² - 12t + 9)dt = [t³ - 6t² + 9t]₀⁴ = (64 - 96 + 36) - 0 = 4 metres. The particle moves 4m from its starting position.

Real-World Application 2: Economics & Consumer Surplus In economics, the area between a demand curve and market price represents consumer surplus—the benefit consumers receive. If demand p = 100 - 2q and equilibrium price is £60, consumer surplus = ∫₀²⁰ (100 - 2q - 60)dq = £400.

Real-World Application 3: Biology & Growth A bacterial culture grows at rate r(t) = 500e^(0.3t) cells/hour. Total growth from hour 2 to hour 5: ∫₂⁵ 500e^(0.3t)dt bacteria.

Analogy: Imagine filling a pool with varying water pressure. The definite integral sums all the tiny amounts of water added each second, giving total volume. Negative regions? That's water draining out! For total water movement (not net), you'd count drainage as positive too—hence using absolute values for area.