I have an image that shows a graph of the function f(x)=x+5e^(-2x) and there is a highlighted square going from 0 to 5 on both the y and x axis, find the area that is under the graph of the function but also inside the highlighted square.

Answer 1

I made a sketch to match your description, and the square intersects the curve at (0,5) and (5,5)
So the effective height of your region is
5 - (x+5e^(-2x) )
= 5 - x - 5e^(-2x)

area = ∫ 5 - x - 5e^(-2x) dx from 0 to 5
= [5x - (1/2)x^2 + (5/2) e^(-2x) ] from 0 to 5
= 25 - 25/2 + (5/2) e^-10 - (0 - 0 + (5/2)e^0 )
= 25/2 + 5/(2e^10) - (5/2)
= 10 + 5/(2e^10 )
= appr 10.00011