posted by anonymous .
f(x)=e^x for 0<x<2. Imagine that you have graphed this function in the x-y plane. For each value of x in the domain, erect a square above the curve perpendicular to the x-axis and parallel to the y-z plane where one side runs from the x-axis to the curve. So the square at x=.9 has vertices (.9,0,0), (.9,e^.9,0), (.9,e^.9,e^.9), (.9,0,e^.9). All of these squares together create a volume. Set up an integral to compute the volume exactly.
length of side of square = e^x
area of square = (e^x)^2 = e^2x
volume = integral dx e^2x from x = 0 to x = 2
= (1/2)e^2x at x = 2 - at x = 0
= (1/2)e^4 -(1/2)(1)
= (1/2)(e^4 -1)