How do you find the volume of the solid generated by revolving the region bounded by the graphs #y=e^(x/2), y=0, x=0, x=4#, about the x axis?

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Answer 1 · 2016-12-15T16:46:23

Please see below.

Here is a graph of the region in blue. A slice has been taken perpendicular to the axis of rotation. The rotation as shown by the arrow/arc.

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The representative slice is a disc of

thickness dx

and radius

#r =y_"greater" - y_"lesser" = e^(x/2)-0 = e^(x/2)# .

The volume of the representative slice (disc) is

#pir^2"thickness" = pi (e^(x/2))^2 dx = pi e^x dx# .

The values of #x# vary from #0# to #4#, so the resulting solid has volume

#V = int_0^4 pi e^x dx#.

Evaluate the integral to get

#V = pi(e^4-1) ~~ 53.6# (cubic units).