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Lesson 5-1 Area Underneath the Curve

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Quiz Homework Problem: Reading questions:

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Objectives Find the area underneath a curve using limits Find the distance traveled by an object (like a car)

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Vocabulary Area problem – find the area under the curve (and the x-axis) between two endpoints Area – is the limit (as n approaches infinity) of the sum of n rectangles Distance problem – find the area under the velocity curve (and the x-axis) between two endpoints

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How do we find the area of the shaded region? a b Area Under the Curve

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Example 1 Sketch the graph and use geometry to find the area: y x 5 2 -2 y x 5 3 2x 0 ≤ x < 1 f(x) = 2 2 ≤ x ≤ 3 5-x 3< x ≤ 5 4 - x² 0 ≤ x ≤ 2 f(x) = x - 2 2 < x ≤ 5 Area = 1 + 4 + 2 = 7Area = ¼ (4π) + ½ (9) = 7.642

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Inscribed Rectangles a b 1 2 3 4 5 The area in yellow is the error in using inscribed rectangles to estimate the area under the curve. Inscribed rectangles underestimate the area! N = 5

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Circumscribed Rectangles a b 12 3 4 5 The area in blue above the curve is the error in using circumscribed rectangles to estimate the area under the curve. Circumscribed rectangles overestimate the area! N = 5

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Inscribed Rectangles a b 1 2 3 4 5 6 7 8 9 10 The area in yellow is the error in using inscribed rectangles to estimate the area under the curve. Inscribed rectangles underestimate the area! N = 10 Less error

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Circumscribed Rectangles a b 1 2 3 4 5 6 7 8 9 10 The area in blue above the curve is the error in using circumscribed rectangles to estimate the area under the curve. Circumscribed rectangles overestimate the area! N = 10 Less error

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Inscribed Rectangles under estimate the area under the curve. Circumscribed Rectangles over estimate the area under the curve. As the number of rectangles increase the error in the estimation decreases. Inscribed vs Circumscribed Summary

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Rectangles from Midpoints The area in yellow is the underestimations and the area in blue are the overestimations of the area. Midpoints seem to give better estimates than either inscribed or circumscribed rectangles. N = 5 a b 1 2 3 4 5

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Trapezoidal Estimates a b The area in yellow is the underestimations and the area in blue are the overestimations of the area. Trapezoids also give better estimates than either inscribed or circumscribed rectangles. N = 5 1 2 3 4 5

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Example 2a Estimate the area bounded by the function f(x) = x² + 1 and the x-axis on the interval [0,2] with 5 subintervals using inscribed rectangles. y x 2 5 0 0 1 2 3 4 5 Area of Rectangle = l ∙ w R 1 = (1+ 0²) ∙ (0.4) = 0.4 R i = (F(x i )) ∙ (∆x) R 2 = (1+ 0.4²) ∙ (0.4) = 0.464 R 3 = (1+ 0.8²) ∙ (0.4) = 0.656 R 4 = (1+ 1.2²) ∙ (0.4) = 0.976 R 5 = (1+ 1.6²) ∙ (0.4) = 1.424 ∑R i = 3.92 x i = a + (i-1)∆x ∆x=(b-a)/n True Area = 4.666

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Summary & Homework Summary: –Inscribed Rectangles under estimate the area under the curve –Circumscribed Rectangles over estimate the area under the curve –Distance traveled is the area under the velocity curve –In order to find the area underneath a curve we must take a limit of the sum of rectangles as the number of rectangles approaches infinity Homework: –pg 378 – 380: 2, 5, 12, 13, 15

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