What is the volume of the solid produced by revolving f(x)=e^x-xlnx, x in [1,5] f(x)=exxlnx,x[1,5]around the x-axis?

1 Answer
Jul 18, 2018

The volume is 29486.55429486.554 "u"^3u3, or the sort of exact result is:

V = pi/2(e^(10) - e^2) - 1696.084pi + (125pi)/3[ln^2(5) - 2/3ln5 + 2/9] - (2pi)/27V=π2(e10e2)1696.084π+125π3[ln2(5)23ln5+29]2π27 "u"^3u3


To do these kinds of solid of revolution problems around the xx axis, just remember that you are projecting a circle whose radius varies as r = f(x)r=f(x).

The volume is then given by:

V = pi int_(a)^(b) f^2(x)dxV=πbaf2(x)dx

The f(x)f(x) serves as the varying radius, and the integral over [a,b][a,b] gives the length of the object, sliced up into discs of thickness dxdx.

In this case, we have:

V = pi int_(1)^(5) (e^x - xlnx)^2dxV=π51(exxlnx)2dx

= pi int_(1)^(5) e^(2x) - 2xe^xlnx + x^2ln^2xdx=π51e2x2xexlnx+x2ln2xdx

= pi overbrace(int_(1)^(5) e^(2x)dx)^("Integral 1") - 2pi overbrace(int_(1)^(5) xe^xlnxdx)^"Integral 2" + pi overbrace(int_(1)^(5) x^2ln^2xdx)^"Integral 3"

  • The first integral is 1/2e^(2x) evaluated from x = 1 to 5, which is 1/2(e^(10) - e^2).
  • The second integral is not possible with elementary functions, and the numerical solution is 848.042.
  • The third integral is hard but doable here.

int x^2ln^2xdx = ?

Let:

  • u = ln^2x
  • dv = x^2dx
  • du = (2lnx)/xdx
  • v = x^3/3

Thus,

int x^2ln^2xdx

= (x^3ln^2x)/3 - int x^3/3 (2lnx)/xdx

= 1/3 x^3ln^2x - 2/3int x^2 lnxdx

Repeat with:

  • u = lnx
  • dv = x^2dx
  • du = 1/xdx
  • v = x^3/3

And we get:

=> 1/3 x^3ln^2x - 2/3[1/3x^3lnx - int x^3/3 1/xdx]

= 1/3 x^3ln^2x - 2/3[1/3x^3lnx - 1/3 int x^2dx]

= 1/3 x^3ln^2x - [2/9x^3lnx - 2/9 int x^2dx]

= 1/3 x^3ln^2x - 2/9x^3lnx + 2/9 int x^2dx

= {:[1/3 x^3ln^2x - 2/9x^3lnx + 2/9 x^3/3]|:}_(1)^(5)

= [1/3 (5)^3ln^2(5) - 2/9(5)^3ln(5) + 2/9(5)^3/3] - [cancel(1/3 (1)^3ln^2(1))^(0) - cancel(2/9(1)^3ln(1))^(0) + 2/9 (1)^3/3]

= [1/3 125ln^2(5) - 2/9(125ln5 - 125/3)] - 2/9(1/3)

= 125/3[ln^2(5) - 2/3ln5 + 2/9] - 2/27

So, adding up all the results,

V = pi { 1/2(e^(10) - e^2)} - 2pi {848.042} + pi {125/3[ln^2(5) - 2/3ln5 + 2/9] - 2/27}

= color(blue)(pi/2(e^(10) - e^2) - 1696.084pi + (125pi)/3[ln^2(5) - 2/3ln5 + 2/9] - (2pi)/27)

or 29486.554.