Question

Given that d/dx(xlnx−x)=lnx, find the volume of the solid obtained by rotating, about the x-axis, the graph of the function y=√(lnx) from the x-intercept and x=e^2.

Answer 1

Since $\frac{d}{dx}(xlnx-x)=lnx,$ then $\int lnx dx= xlnx-x+C$

Volume of the solid obtained by rotating can be found by formula: $\pi \int y^2dx$

Boundaries of integration are $x=1$ and $x=e^2$.

So:

$V=\pi \int\limits_1^{e^2} (\sqrt{lnx})^2dx=\pi\int\limits_1^{e^2}lnxdx=\pi (xlnx-x)\bigg|_1^{e^2}=\\ \pi(e^2lne^2-e^2)-\pi(1ln1-1)=\pi(e^2+1)$