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MathCalculus

Second Fundamental Theorem of Calculus: Approximate and Exact Area under y = x²

Compute L₁₀, R₁₀ and their average for y=x² on [0,4], then use the second fundamental theorem of calculus to obtain the exact area.

Question

For y=x2y=x^2 on [0,4][0,4], use a calculator with N=10N=10 equal-width rectangles to compute the left- and right-endpoint Riemann sums. Estimate the area using T10T_{10}, their average. Then determine the exact area using the Fundamental Theorem of Calculus, Part 2.

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Evidence boundary

This page answers both requests in OpenStax Calculus Volume 2, §1.3, exercise 164: the average endpoint estimate with ten equal-width rectangles and the exact integral. The displayed numerical sums are reproducible arithmetic, not experimental observations. The integral is area here because x² is nonnegative throughout the stated interval.