Fundamental Theorem of Calculus

Business Calculus · Axiom Academy

LESSON The Fundamental Theorem of Calculus The bridge connecting differentiation and integration The Fundamental Theorem of Calculus (FTC) is arguably the most important theorem in calculus. It establishes that: Differentiation and Integration are inverse operations Without the FTC, we'd have to compute integrals using limits of Riemann sums - extremely tedious! The FTC gives us a shortcut: find an antiderivative and evaluate at the endpoints. Part 1: Differentiation of Integrals The derivative of an integral (with variable upper limit) gives back the original function. Part 2: Evaluation of Integrals Definite integrals can be evaluated using any antiderivative. Part 1: "Integration and differentiation undo each other" Part 2: "To find the area under a curve, find an antiderivative and subtract endpoint values" If f is continuous on [a, b] and we define: Then F is differentiable on (a, b) and: By FTC Part 1, the derivative equals the integrand evaluated at x: Example with Chain Rule: dx _0^ x^2 t^3\,dt When the upper limit is a function of x, use the chain rule: FTC Part 2 (Evaluation Theorem) If f is continuous on [a, b] and F is any antiderivative of f, then: Example: Evaluate _1^4 (3x^2 - 2x + 1)\,dx The FTC is the mathematical foundation for calculating totals from rates: Cost of producing units 50 through 100 Profit accumulated from year 0 to year 5 The integral of a rate of change gives the total change

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