The area under the curve y=e^-x between x=0 and x=1 is rotated about the x axis find the volume?

1 Answer
Apr 10, 2018

Answer:

Volume is #pi/2(1-e^-2)=1.358# cubic units.

Explanation:

Let us see the graph of #y=e^(-x)# between #x=0# and #x=1#.

graph{e^(-x) [-2.083, 2.917, -0.85, 1.65]}

To find the desired volume the shaded portion (shown below, will have to be rotated around #x#-axis.

As volume of a cylinder is #pir^2h#, here we will have #r=e^(-x)# and #h=dx#

and hence volume would be

#int_0^1pie^(-2x)dx#

= #piint_0^1e^(-2x)dx#

= #pixx[-e^(-2x)/2]_0^1#

= #pi[-e^(-2)/2+1/2]#

= #pi/2(1-e^-2)#

= #pi/2(1-0.1353)#

= #1.358#