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Unformatted text preview: Area between curves Definition : Let y 1 = f 1 ( x ) and y 2 = f 2 ( x ) , x [ a, b ] , be two curves such that f 1 ( x ) > f 2 ( x ) for all x [ a , b ] . Let us then define y upper = f 1 ( x ) and y lower = f 2 ( x ). The area of the region bounded by the curves f 1 ( x ) and f 2 ( x ) and the vertical lines x = a and x = b is given by: Area = b Z a ( y upper- y lower ) dx = b Z a ( f 1 ( x )- f 2 ( x )) dx, (1) where y upper = y upper ( x ) is the curve higher (graphically) and y lower = y lower ( x ) is the curve lower (graphically), for all x [ a, b ]. Note 1 : y upper ( x ) and y lower ( x ) do not intersect for all x ( a, b ). Note 2 : The area is always a positive number. Special Cases : y upper = x-axis or y lower = x-axis. The equation of the x-axis is y = y ( x ) = 0. There are 2 cases to consider. Case 1 : Let y = f ( x ) be a curve such that all y-values are positive for all x [ a, b ], i.e. f ( x ) 0 for all x [ a , b ]. Then y upper = f ( x ) and y lower = x-axis, i.e. y lower = 0. The area between the curve y = f ( x ), the x-axis, the vertical lines x = a and x = b is thus: Area (above the x-axis) = b Z a ( y upper- y lower ) dx = b Z a [ f ( x )- 0] dx, by (1) i.e. Area (above the x-axis) = b Z a f ( x ) dx. (2) Example (Ex 6, Sect 4.7) Evaluate the area of the region bounded by y = 3 x 2 , the x-axis and the lines x = 1 and x = 3 . Solution All the y s on the curve y = 3 x 2 are positive for all x [1 , 3] . Since equation of x-axis is y ( x ) = y = 0 and thus y lower = 0, the area of the given region is simply: Area = 3...
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- Spring '11