5_1 - Section 5.1: Areas and Distances Instructor: Ms. Hoa...

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Section 5.1: Areas and Distances Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu) 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 (where f ( x ) 0), the vertical lines x = a and x = b , and the x-axis. Example 1 : Find the area under the parabola y = x 2 from 0 to 1. First, we approximate the region S by rectangles and then we take the limit of the areas of these rectangles as we increase the number of rectangles. Divide S into 4 strips, and approximate each strip by a rectangle. The height of each rectange is the value of f at the RIGHT endpoint of each subinterval. The sum of the areas of these four rectangles is: R 4 = 1 4 · ( 1 4 ) 2 + 1 4 · ( 1 2 ) 2 + 1 4 · ( 3 4 ) 2 + 1 4 · (1) 2 = 15 32 = 0 . 46875 Instead of using the above rectangles, we use the smaller rectangles. Divide S into 4 strips, and approximate each strip by a rectangle. The height of each rectange is the value of
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5_1 - Section 5.1: Areas and Distances Instructor: Ms. Hoa...

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