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Applications of the Integral

Problems

Length of a Graph

Distance Traveled

Problem : Find the length of the graph of f (x) = (1/2)(e x + e -x) from x = 0 to b .

Since f'(x) = (1/2)(e x - e -x) , the length of the graph is equal to


dx = e x + e -x dx  
  = (e x - e -x|0 b)  
  =  

Problem : Find the length of the graph of g(x) = 2x 3/2 from x = 2 and x = 5 .

Since g'(x) = 3 , the length is given by dx . Making the change of variables u = 1 + 9x , we have


9 dx = u 1/2 du  
  = |19 46  
  = (463/2 -193/2)  

or approximately 16.98 .

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