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CHAPTER 2 2.4 Continuity Areas and Distances

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The area problem: Find the area of the region S that lies under the curve y = f (x) from a to b. This means that S, is bounded by the graph of a continuous function f, the vertical lines x=a and x=b, and the x-axis.

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Example Estimate the area under the graph of f (x) = x 3 +2 from x = -1 to x=2 using three rectangles and right endpoints. Sketch the curve and the approximating rectangles.

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CHAPTER 2 2.4 Continuity animation f (x) xx xixi x i+1

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Definition The area A of the region S lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: A = lim n 00 R n = lim n 00 [ f (x 1 ) x + … + f (x n ) x ]

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Example Use definition to find an expression for the area under the curve y = x 3 from 0 to 1 as a limit.

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The Distance Problem: Find the distance traveled by an object during a certain time period if the velocity of the object is known at all times. If the velocity remains constant, then the distance problem is easy to solve by means of the formula : distance = velocity x time.

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t (s)00.51.01.52.02.53.0 v (f t / s)06.210.814.918.119.420.2 Example The speed of a runner increased by steadily the first 3 sec. of a race. Her speed at ½ during second intervals are given in the table. Find lower and upper estimates for the distance that she traveled during these 3 sec.

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