find the volume of the solid generated by revolving the regi

Solution

V = integral of [f(x)]^2 dx.....limit from 1 to 4
V = integral of (lnx)^2dx
let t = lnx
dt = 1/xdx
dx = xdt
dx = e^tdt
V = integral of e^t*t^2dt
apply by parts
V = [t^2 integral of e^t - integral of (d/dt(t^2) integral of e^t]
V = [t^2 e^t - integral of {2te^t}]
V = [t^2e^t - {2tintegral of e^t - 2integral of (d/dt(t) integral of e^t)}]
V = [t^2e^t - {2te^t - integral of 2e^t}]
V = [t^2e^t - {2te^t - 2e^t}]
V = [t^2e^t - 2te^t + 2e^t]
since e^t = x
V = [x(lnx)^2 - 2xlnx + 2x]
apply limits
V = [4(ln4)^2 - 8ln4 + 8 - 0 + 0 - 2]
V = [16(ln2)^2 - 16ln2 + 6]