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Find the area?

Find the area enclosed by the curves y=x^3-8x^2+18x-5 and y=x+5. 
 
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Richard P. | Fairfax County Tutor for HS Math and ScienceFairfax County Tutor for HS Math and Sci...
4.9 4.9 (587 lesson ratings) (587)
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This problem makes sense if the domain of the two functions is something like [0.3].  Then the area enclosed by the two curves is a small region specified by the two values of x for which the functions have the same value.  These two values are x =1 and x =2.
On the interval  (1,2)   the cubic function is greater than the linear one.   Therefore
 
Area = integral from 1 to 2 of x3 -8 x2 +18 x -5 -   (x +5) =
       integral  from 1 to 2    of   x3 -8 x2 + 17 x -10
 
The antiderivative can be found term by term 
 
  [ (1/4) x4 - (8/3) x3 + (17/2) x2 - 10x ]  
 
area =   (this expression evaluated at x = 2)   -  (this expression evaluated at x =1 )
area = 7/12
 
 

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