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Find its volume?

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
 
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2 Answers

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 =∫03 e-2x dx = [-(1/2) e-2x]03 = -(1/2) (e-6 - 1) = (1 - e-6)/2.

Comments

 
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-2xdx 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 - e0] = -1/2 [ e-6 -1] = +1/2[1 - e-6
 
[1 - e-6]/2

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