Find its volume?

Question

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.

Answer 1

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.

Answer 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

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