Calculus 1 — 22.1: The Rectangular Method for Area Under a Curve

Описание: How do you find the area of a shape with no formula? By filling it with rectangles. This video walks through the rectangular method step by step, using f(x) = x² + 1 on the interval [0, 3] to show how summing rectangle areas approximates the area under a curve — and how increasing the number of rectangles drives the approximation toward the exact answer.

Key concepts covered:
• Why curved regions require a new approach beyond simple geometry formulas
• The two fundamental problems of calculus: slopes (derivatives) and areas (integrals)
• Dividing an interval into n equal subintervals and computing Δx = (b − a) / n
• Left-endpoint, right-endpoint, and midpoint rectangle methods
• Full step-by-step left-endpoint computation with n = 6, yielding an approximation of 9.875
• How increasing n from 6 to 20 to 100 rectangles reduces the error from 2.125 to 0.13
• The limit concept: as n approaches infinity and Δx shrinks to zero, the sum becomes the exact area
• The connection between the Δx → 0 limit in derivatives and the same limit in area approximation
• Why left endpoints underestimate and right endpoints overestimate on an increasing function
• A six-step toolkit for approximating area under any curve
• Preview of the antiderivative as the shortcut for computing exact areas

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SOURCE MATERIALS
The source materials for this video are from    • Calculus 1 Lecture 4.1:  An Introduction t...