Chap6_Sec1

Chap6_Sec1 - AREAS BETWEEN CURVES Example 1 (6.1) Find the...

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Find the area of the region bounded above by y = e x , bounded below by y = x, and bounded on the sides by x = 0 and x = 1. As shown here, the upper boundary curve is y = e x and the lower boundary curve is y = x . So, we use the area formula with y = e x , g ( x ) = x , a = 0, and b = 1: AREAS BETWEEN CURVES Example 1 (6.1) ( ) 1 1 2 1 2 0 0 1 1 1.5 2 x x A e x dx e x e e = - = - = - - = -
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Find the area of the region enclosed by the parabolas y = x 2 and y = 2x-x 2 . First, we find the points of intersection of the parabolas by solving their equations simultaneously. This gives x 2 = 2 x - x 2 , or 2 x 2 - 2 x = 0. Thus, 2 x ( x - 1) = 0, so x = 0 or 1. The points of intersection are (0, 0) and (1, 1). From the figure, we see that the top and bottom boundaries are: y T = 2 x x 2 and y B = x 2. The area of a typical rectangle is ( y T y B ) x = (2 x x 2 x 2 ) x and the region lies between x = 0 and x = 1. So, the total area is:
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This note was uploaded on 05/23/2011 for the course MAC 2311 taught by Professor Noohi during the Fall '08 term at FSU.

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Chap6_Sec1 - AREAS BETWEEN CURVES Example 1 (6.1) Find the...

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