Question

find area  enclosed by the given curve the x-axis and the given co- ordinates  x=0 and x=2 for the functions y=e^2x - e^3x

Answer 1

The area is given by integral(ydx) (0<x<2)=integral((e^2x-e^3x)dx)=[(1/2)e^2x-(1/3)e^3x](2,0)=((1/2)e^4-(1/3)e^6-(1/2)+1/3)=(-107.177-1/6)=-107.344. The minus sign means the area is below the x axis, so the actual size of the area is 107.344 approx. The function is zero at x=0 and is negative thereafter, so the whole area is below the x axis.