^{-x}, and, since it is a square, height also e

^{-x}:

^{-x})(e

^{-x}) dx = e

^{-2x}dx

_{0}

^{3}e

^{-2x}dx = [-(1/2) e

^{-2x}]

_{0}

^{3}= -(1/2) (e

^{-6}- 1) = (1 - e

^{-6})/2.

The base of a solid is the region enclosed by the graph of y=e^-x, the coordinate axes, and the line x=3. If all plane cross sections perpendicular to the x-axis are squares, find its volume.

Answer: (1-e^-6)/2

Please show all your work.

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Find the infinitesimal volume of one cross-sectional slice of thickness dx, width y=e^{-x}, and, since it is a square, height also e^{-x} :

dV = (e^{-x})(e^{-x}) dx = e^{-2x} dx

Integrate from 0 to 3 to find the total volume:

V =∫_{0}^{3} e^{-2x} dx = [-(1/2) e^{-2x}]_{0}^{3} = -(1/2) (e^{-6} - 1) = (1 - e^{-6})/2.

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

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}]

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