5.1.1-eg - f c 1 Δ x f c 2 Δ x f c 3 Δ x =(1 4 5 ...

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Example Estimate the area under the graph of y = f ( x ) = 5 - x 2 from x = - 2 to x = 1 using the areas of 3 rectangles of equal widths, with heights of the rectangles determined by the height of the curve at (a) left endpoints, and (b) right endpoints. Solution: The exact area in question is shaded below. - 2 - 1 1 2 - 1 1 2 3 4 5 The width of the equally-spaced rectangles is given by Δ x = b - a n where in this case, a = - 2, b = 1, and n = 3, so Δ x = 1 - ( - 2) 3 = 1 . (a) Left endpoints: In this case, the left endpoints of the intervals are given by c k = a + ( k - 1)Δ x = - 2 + ( k - 1) , k = 1 , 2 , 3 , or c 1 = - 2 , c 2 = - 1 , c 3 = 0 . - 2 - 1 1 2 - 1 1 2 3 4 5 The heights of the rectangles are given in the following table:
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k c k f ( c k ) 1 - 2 f ( - 2) = 5 - ( - 2) 2 = 1 2 - 1 f ( - 1) = 5 - ( - 1) 2 = 4 3 0 f (0) = 5 - (0) 2 = 5 So the approximate area is given by
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Unformatted text preview: f ( c 1 ) · Δ x + f ( c 2 ) · Δ x + f ( c 3 ) · Δ x = (1 + 4 + 5) · 1 = 10 . (b) Right endpoints: The right endpoints of the intervals are given by c k = a + k Δ x =-2 + k, k = 1 , 2 , 3 , or c 1 =-1 , c 2 = 0 , c 3 = 1 .-2-1 1 2-1 1 2 3 4 5 The heights of the rectangles are given in the following table: k c k f ( c k ) 1-1 f (-1) = 5-(-1) 2 = 4 2 f (0) = 5-(0) 2 = 5 3 1 f (1) = 5-1 2 = 4 So the approximate area is given by f ( c 1 ) · Δ x + f ( c 2 ) · Δ x + f ( c 3 ) · Δ x = (4 + 5 + 4) · 1 = 13 . 2...
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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5.1.1-eg - f c 1 Δ x f c 2 Δ x f c 3 Δ x =(1 4 5 ...

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