How do you find the volume of the solid generated when the regions bounded by the graphs of the given equations #y=e^-x#, x= -1, x = 2 and the x-axis are rotated about the x-axis?

Question

Isabella Ackerman · Accepted Answer

# VOR = pi/2 (e^2-e^(-4)) #
# " " ~~11.5779320... #

I recommend that you always draw a sketch to clarify what needs calculating. graph{e^(-x) [-2, 3, -5, 5]} The Volume of Revolution about #Ox# is given by: # VOR = int_alpha^beta \ piy^2 \ dx # So in this case: # VOR = int_-1^2 \ pi \ (e^-x)^2 \ dx # # " " = pi \ int_-1^2 \ e^(-2x) \ dx # # " " = pi [ -1/(2) e^(-2x) ]_-1^2 # # " " = -pi/2 {(e^(-4))-(e^2)} # # " " = pi/2 (e^2-e^(-4)) # # " " ~~11.5779320... #

Isabella Ackerman · Answer

undefined To find the volume of the solid generated by rotating the region bounded by the graphs of the equations $y = e^{-x}$, $x = -1$, $x = 2$, and the x-axis about the x-axis, you use the method of cylindrical shells. The volume can be calculated using the following integral:

$$
V = \int_{-1}^{2} 2\pi x \cdot e^{-x} \, dx
$$

Where $V$ represents the volume of the solid, $x$ ranges from $-1$ to $2$, and $2\pi x$ represents the circumference of the cylindrical shell at each $x$-value. $e^{-x}$ represents the height of each shell. Integrating this expression will give the volume of the solid.