123-1-2510152025xyOpen image in a new page

The graph of `y=e^x`, with the area under the curve between `x=0` to `x=3` shaded.

When the shaded area is rotated 360° about the x-axis, we have:

xOpen image in a new page

Area under the curve `y=e^x` from `x=0` to `x=3` rotated around the `x`-axis.

Applying the volume of a solid of revolution formula, we get

`V=pi int_a^by^2dx`

`=pi int_0^3(e^x)^2dx`

`=pi int_0^3e^(2x)dx`

`=pi[e^(2x)/2]_0^3`

`=pi[(e^6)/2]-pi[(1)/2]`

`=((e^6-1)/2)pi\ "units"^3`

`=632.1\ "units"^3`