The height is given by the equation y = e^{-x}. The area of a square is given by x ·y. Since the cross-sections are squares, the height and width will be the same. So the area of each individual square is (e^{-x})^{2} = e^{-2x}.
To find the volume, add up all the infinitely thin squares from x = 0 to x = 3. The equation looks like
V = ∫e^{-2x}dx where the limits of integration are x = 0 to x = 3.
Integrating, we get -(1/2) e^{-2x}. Now if we use the limits, -1/2 [ e^{-6} - e^{0}] = -1/2 [ e^{-6} -1] = +1/2[1 - e^{-6}]
[1 - e^{-6}]/2
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