Question

What is the volume of the solid produced by revolving `f(x)=xe^x-(x/2)e^x, x in [2,7]`around the x-axis?

Answer 1

The volume of the solid produced by revolving around the x-axis the trapezoid of the function `f(x)` is calculated using the formula:

`V = pi int_a^b f^2(x)dx`

Explanation:

`V= pi int_2^7 (xe^x-(x/2)e^x)^2dx`

`V= pi int_2^7 (x^2e^(2x)-2xe^x(x/2)e^x+x^2/4e^(2x))dx`

`V= pi int_2^7 (x^2e^(2x)-x^2e^(2x)+x^2/4e^(2x))dx = pi int_2^7 x^2/4e^(2x)dx`

Substitute `2x=t`:

`V = pi/32 int_1^(7/2) t^2e^tdt`

The integral is solved iteratively by parts:

`int t^2e^tdt = int t^2d(e^t) =t^2e^t - int 2te^tdt = t^2e^t - 2te^t +2inte^tdt = e^t(t^2-2t+2)`

Finally:

`V = pi/32[e^(7/2)(49/4-7+2)-e(1-2+2)] = pi/32[e^(7/2)(49-28+8)/4-e] = (pi*e)/32 (29/4e^(5/2)-1)`