Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. `y = e^(x/2) , y = 0 , x = 0 , x = 4`
Sketch the region bounded by the graphs of the given equations.
`y=e^(x/2) ` (Blue curve)
`y=0` (Yellow line)
`x=0` (Red line)
`x=4` (Green line)
To solve for the volume of the solid formed by revolving the bounded region around the x-axis, apply the disk method. The formula when the axis of rotation is horizontal is:
`V=int_a^b pi r^2 dx`
In our graph above, the radius is the distance between the axis of rotation and the blue curve.
And the limits of the integral are the end x values of the bounded region.
Plugging them to the formula of volume, we would have:
`V=int_0^4 pi (e^(x/2))^2 dx`
`V=pi int_0^4 e^(x/2*2) dx`
`V=pi int_0^4 e^x dx`
Evaluating the integral yields:
`V= pi (e^x)|_0^4`
Therefore, the volume of the solid formed by revolving the bounded region around the x-axis is 168.4 cubic units.