Lessons 25–26 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 11, Parts 1 and 2, 1st edition. ZAMIN NASHR, Tashkent, 2018
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Approximate integration
Textbook, Part 2: pp. 10–12
GoalLearn to approximate a definite integral with the rectangle and trapezoid formulas.
Many functions have no antiderivative in elementary form, so we approximate the definite integral. Split [a, b] into n equal parts: h = (b − a)/n, with division points xₖ = a + kh. The left rectangle rule is ∫ₐᵇ f dx ≈ h · (y₀ + y₁ + … + yₙ₋₁) and the right rectangle rule is h · (y₁ + … + yₙ), where yₖ = f(xₖ). The midpoint rule uses the value at the middle of each part. The trapezoid rule is ∫ₐᵇ f dx ≈ h · (y₀/2 + y₁ + … + yₙ₋₁ + yₙ/2); it equals the average of the left and right sums. For an increasing function the left sum underestimates and the right sum overestimates; the larger n is, the better the accuracy.
Worked examples
Estimate ∫₀¹ x² dx with n = 4: h = 0.25, y = 0; 0.0625; 0.25; 0.5625; 1. Left: 0.25 · 0.875 = 0.21875; right: 0.25 · 1.875 = 0.46875; trapezoid: the average, 0.34375. The exact value is 1/3 ≈ 0.3333, so the trapezoid error is about 0.01.
We compute ∫₀² x³ dx (exact value 4) by the trapezoid rule with n = 4: h = 0.5; y = 0; 0.125; 1; 3.375; 8. The sum is 0/2 + 0.125 + 1 + 3.375 + 8/2 = 8.5; the result is 0.5 · 8.5 = 4.25. The function is convex, so the trapezoids overestimate the integral a little.
Class activity
“Integral from a table”: velocity readings are given: at t = 0, 1, 2, 3, 4 s the speed is v = 0; 3; 5; 6; 6 m/s. With the trapezoid rule find the distance covered in 4 seconds (∫v dt) and compare it with the rectangle result.
Practice
1
Find ∫₀² x² dx with the left rectangle rule for n = 2.
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2
Find ∫₀² x² dx with the trapezoid rule for n = 2.
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Find ∫₀³ x dx with the right rectangle rule for n = 3.
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4
Why is the left sum smaller than the integral for an increasing function?
On each part the value at the left end is the smallest, so the rectangle lies below the graph and its area is less than that of the curvilinear trapezoid.