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# Area - AREAS AND DISTANCES Suppose that we want to find the...

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AREAS AND DISTANCES Suppose that we want to find the area under a curve. First of all, we need to define what the area is. If we have a rectangle, it is relatively easy, because we can simply define the area as the product of the length and the width. We will use this property of a rectangle as a base point of defining the area under a curve. Example 1. Use rectangles to estimate the area under the parabola y = x 2 from 0 to 1. Solution Suppose we divide the parabolic region into four strips. We can ap- proximate each strip by a rectangle whose base is the same as the strip and whose height is the same as the right edge of the strip. Each rectangle has width 1 4 and the heights are ( 1 4 ) 2 , ( 2 4 ) 2 , ( 3 4 ) 2 , and 1 2 . If we let R 4 be the sum of the areas of these rectangles, we get R 4 = 1 4 ( 1 4 ) 2 + 1 4 ( 2 4 ) 2 + 1 4 ( 3 4 ) 2 + 1 4 · 1 2 = 15 32 = 0 . 46875 We can also see that the area A of the parabolic region is less than R 4 , so A < 0 . 46875 Alternatively, we can use smaller rectangle whose heights are the values of f at the endpoints of the subintervals. The sum of these approximating rectangles is L 4 = 1 4 · 0 2 + 1 4 ( 1 4 ) 2 + 1 4 ( 2 4 ) 2 + 1 4 ( 3 4 ) 2 = 7 32 = 0 . 21875 Note that the leftmost rectangle has collapsed.

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Area - AREAS AND DISTANCES Suppose that we want to find the...

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