Evaluating ∫₁⁴ (2x + 1) dx

Real Analysis · Axiom Academy

Applying the Fundamental Theorem of Calculus to evaluate a definite integral, step by step. Evaluate the definite integral using the Fundamental Theorem of Calculus (Part 2). The integral is the area of the trapezoid under y = 2x + 1 from x = 1 to x = 4: heights 3 and 9, width 3, so area = ((3 + 9) / 2) × 3 = 18. Nice work! You evaluated a definite integral with the Fundamental Theorem of Calculus. The essential moves: Identify the integrand and limits: here, integrate 2x + 1 over the interval [1, 4] . Find an antiderivative: F(x) = x^2 + x , since F'(x) = 2x + 1 . Evaluate at the endpoints: F(4) = 20 and F(1) = 2 . Subtract for the result: 20 - 2 = 18 . The FTC turns an area problem into evaluating an antiderivative at two points. Geometrically, is the area of the trapezoid under the line y = 2x + 1 on [1, 4] — width 3 , heights 3 and 9 , so , the same answer.

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