SOLUTION: Integration :Find the area bounded by y=xe^-x, y=0, and x=2

The first step is to note that the lower limit of integration will be x=0.  This is because y=0 is one of the boundaries and y is negative for x<0.  So we will have limits of integration of x=0 and x=2.



Use integration by parts:



   Let          and    

giving us    and     





Recall integration by parts:

 =  - 



 =  - 

                        =  - 

                        = 



Evaluate this expression at x=2:  -0.4060  (approx)

                     and at x=0:   -1



Subtract the bottom from the top:  -0.4060-(-1) = 0.5940 sq units. (to 4 decimal places)