`y = xe^(-x), y = 0, x = 2` (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve...

The shell has the radius x, the cricumference is `2pi*x` and the height is `x*e^(-x)` , hence, the volume can be evaluated, using the method of cylindrical shells, such that:

`V = 2pi*int_(x_1)^(x_2) x*x*e^(-x) dx`

You need to find the next endpoint, using the equation `x*e^(-x) = 0 => x = 0`

`V = 2pi*int_0^2 x^2*e^(-x) dx`

You need to use integration by parts to evaluate the volume, such that:

`int udv = uv - int vdu`

`u = x^2 => du = 2xdx`

`dv = e^(-x) => v = -e^(-x)`

`int_0^2 x^2*e^(-x) dx = -x^2*e^(-x)|_0^2 + 2int_0^2 x*e^(-x)dx`

You need to use integration by parts to evaluate the integral `int_0^2 x*e^(-x)dx.`

`u = x => du = dx`

`dv = e^(-x) => v = -e^(-x)`

`int_0^2 x*e^(-x)dx = -x*e^(-x)|_0^2 + int_0^2 e^(-x) dx`

`int_0^2 x*e^(-x)dx = -x*e^(-x)|_0^2 - e^(-x)|_0^2`

`int_0^2 x*e^(-x)dx = -2*e^(-2) - e^(-2) +0*e^(0)+ e^(0)`

`int_0^2 x*e^(-x)dx = -2/(e^2) - 1/(e^2) + 1`

`int_0^2 x*e^(-x)dx = -3/(e^2)+ 1`

`int_0^2 x^2*e^(-x) dx = -x^2*e^(-x)|_0^2 + 2(-3/(e^2)+ 1)`

`int_0^2 x^2*e^(-x) dx = -2^2*e^(-2) - 6/(e^2) + 2`

`int_0^2 x^2*e^(-x) dx = -4/(e^2) -6/(e^2) + 2`

`int_0^2 x^2*e^(-x) dx = -10/(e^2) + 2`

`V = 2pi*(-10/(e^2) + 2)`

Hence, evaluating the volume, using the method of cylindrical shells, yields `V = 2pi*(-10/(e^2) + 2).`