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Applications of the Integral

Problems

Areas in the Plane

Volumes of Solids

Problem : Find the area of the region in the plane between the graphs of f (x) = | x| and g(x) = - x 2 + 6 .

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


(- x 2 + 6 - | x|)dx = (- x 2 +6)dx - (- x)dx - xdx  
  = + 6x - |-2 0 - |0 2  
  =  

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

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


| e x - (1 + 2x)| dx = (e x -1 - 2x)dx + (1 + 2x - e x)dx  
  = [e x - x - x 2]-1 0 + [x + x 2 - e x]0 1  
  = 1 - + [(2 - e) - 1]  
  =  

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