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Integral as Area
Riemann Sums, the Fundamental Theorem, and Accumulation
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The definite integral ∫[a to b] f(x) dx measures the signed area between the curve f(x) and the x-axis from x = a to x = b. The concept arises from Riemann sums: dividing the interval [a, b] into n subintervals, forming rectangles of width Δx and height f(xᵢ), and summing all their areas. As n → ∞ and Δx → 0, this sum converges to the exact integral. Regions where f(x) is negative contribute negative area to the definite integral.
The Fundamental Theorem of Calculus (FTC) connects differentiation and integration. Part 1 states that if F is the antiderivative of f, then d/dx[∫f(t)dt from a to x] = f(x). Part 2 provides the evaluation shortcut: ∫[a to b] f(x) dx = F(b) − F(a), where F is any antiderivative of f. Integration also computes displacement, net change, work, and probability — any quantity that accumulates continuously over a continuous variable. The indefinite integral ∫f(x) dx = F(x) + C finds families of antiderivatives, with the constant C representing the entire family.
Key Facts
- Definite integral = signed area under curve; negative where f(x) < 0
- Riemann sum: Σ f(xᵢ)Δx → ∫[a to b] f(x) dx as n → ∞
- FTC Part 2: ∫[a to b] f(x) dx = F(b) − F(a), where F'(x) = f(x)
- Power rule for integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
- ∫[a to b] f(x) dx = −∫[b to a] f(x) dx (reversing limits changes sign)
- Net displacement ≠ total distance; total distance = ∫|v(t)| dt
Vocabulary
- Definite integral
- The limit of a Riemann sum giving the signed area under a curve f(x) between x = a and x = b.
- Antiderivative
- A function F(x) whose derivative is f(x); also called an indefinite integral. Differs from another antiderivative by only a constant C.
- Riemann sum
- An approximation of the area under a curve using rectangles; the basis for the definition of the definite integral.
- Fundamental Theorem of Calculus
- The theorem connecting differentiation and integration: differentiation and integration are inverse operations, and antiderivatives evaluate definite integrals.
- Accumulation function
- A function of the form A(x) = ∫[a to x] f(t) dt that represents the accumulated area (or net change) from a fixed point a up to x.
Common Mistakes to Avoid
- Forgetting the constant of integration for indefinite integrals. ∫f(x) dx = F(x) + C, not just F(x). The constant matters because it represents an entire family of antiderivatives.
- Treating negative areas as zero. If f(x) is negative on part of [a,b], the integral subtracts that region's area. The definite integral gives net signed area, not total area.
- Applying the power rule to ∫(1/x) dx. ∫xⁿ dx = xⁿ⁺¹/(n+1) fails when n = −1 (division by zero). Instead, ∫(1/x) dx = ln|x| + C.
- Confusing indefinite and definite integrals. ∫f(x) dx is a family of functions + C; ∫[a to b] f(x) dx is a single number (the area). The definite integral needs bounds.
Practice Questions
- 1 Evaluate ∫[0 to 3] (x² − 2x) dx. Identify any regions where the integrand is negative and how they affect the result.
- 2 Find the total area (not net area) between f(x) = x² − 4 and the x-axis from x = 0 to x = 3.
- 3 A particle moves with velocity v(t) = t² − 4t + 3 m/s. Find the displacement and total distance traveled from t = 0 to t = 4 s.