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References: Riemann Sums and the Fundamental Theorem

  1. Riemann sum - Wikipedia - Left, right, midpoint, and trapezoidal Riemann sums with convergence to the definite integral. Core reference for the approximation methods in this chapter.

  2. Fundamental theorem of calculus - Wikipedia - Both parts of the FTC with proofs and geometric interpretations. The central theorem connecting differentiation and integration presented in this chapter.

  3. Riemann integral - Wikipedia - Formal definition of the definite integral as a limit of Riemann sums. Provides the rigorous foundation for the area-under-curve concept developed in this chapter.

  4. Calculus: Early Transcendentals (9th Edition) - James Stewart - Cengage Learning - Sections 5.1-5.3 develop Riemann sums, the definite integral, and both parts of the Fundamental Theorem with clear geometric motivation.

  5. Thomas' Calculus (15th Edition) - Joel Hass, Christopher Heil, Maurice Weir - Pearson - Sections 5.1-5.4 provide excellent visualizations of Riemann sums converging to the integral and careful treatment of both FTC parts.

  6. Riemann Sums - Paul's Online Math Notes - Step-by-step construction of Riemann sums with worked examples showing convergence as the number of rectangles increases.

  7. Fundamental Theorem of Calculus - Khan Academy - Interactive lessons on both FTC parts with practice evaluating definite integrals and finding derivatives of accumulation functions.

  8. Essence of Calculus: Integration - 3Blue1Brown - Visually stunning explanation of how area accumulation connects to antiderivatives, making the FTC feel intuitive rather than magical.

  9. Riemann Sums Visualization - Desmos - Interactive Riemann sum visualizer where students can adjust the number of rectangles and see convergence to the definite integral.

  10. The Fundamental Theorem of Calculus - Whitman College Calculus - Open-source section with clear proofs of both FTC parts and exercises connecting area functions to antiderivatives.