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Problems

Problem : Find the area of the region in the plane between the graphs of f (x) = | x| and g(x) = - x2 + 6.

The intersection points of the graphs are at x = ±2, and - x2 +6≥| x| on the interval [- 2, 2], so the area of the relevant region is equal to


(- x2 + 6 - | x|)dx=(- x2 +6)dx - (- x)dx - xdx  
 = + 6x - |-20 - |02  
 =  

Problem : Find the total area between the graphs of f (x) = ex and g(x) = 1 + 2x from x = - 1 to x = 1.

Note that ex≥1 + 2x on the interval [- 1, 0] and 1 + 2xex on the interval [0, 1]. Therefore the total area between the graphs is


| ex - (1 + 2x)| dx=(ex -1 - 2x)dx + (1 + 2x - ex)dx  
 =[ex - x - x2]-10 + [x + x2 - ex]01  
 =1 - + [(2 - e) - 1]  
 =