Find the area?

Question

Find the area enclosed by the curves y=x^3-8x^2+18x-5 and y=x+5. 

 

Please show all your work.

Answer 1

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