`y = x, y = xe^(1 - (x/2))` Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the...

These curves are intersected at `x=0` and `x=2.`

Between these points `0lt=xlt=xe^(1-x/2)lt=2.`

Let's use the method of rings.

The parameter of a ring is `x` between `0` and `2.`

The area of a ring is `pi[(3-x)^2-(3-xe^(1-x/2))^2].`

Therefore the volume is equal to

`pi int_0^2[(3-x)^2-(3-xe^(1-x/2))^2] dx.`

Computer algebra system WolframAlpha says that the exact value is `(2pi)/3 (36e-3e^2-71).`

These curves are intersected at `x=0` and `x=2.`

Between these points `0lt=xlt=xe^(1-x/2)lt=2.`

Let's use the method of rings.

The parameter of a ring is `x` between `0` and `2.`

The area of a ring is `pi[(3-x)^2-(3-xe^(1-x/2))^2].`

Therefore the volume is equal to

`pi int_0^2[(3-x)^2-(3-xe^(1-x/2))^2] dx.`

Computer algebra system WolframAlpha says that the exact value is `(2pi)/3 (36e-3e^2-71).`