Question

find out volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=lnx, y=1, y=3,x=0; about the y-axis.

Answer 1

$Look\ at\ the\ picture.\\\\y=lnx\\\\x=lny\to e^x=e^{lny}\to y=e^x\\\\V=\int\limits_1^3\left(e^x\right)^2dx=(*)\\\\\int\left(e^x\right)^2dx=\int(e^x\times e^x)dx\Rightarrow \left|\begin{array}{ccc}e^x=t\\\\e^x\ dx=dt\end{array}\right|\Rightarrow\int\limits_1^3t\ dt=\frac{1}{2}t^2=\frac{1}{2}\left(e^x\right)^2=\frac{e^{2x}}{2}$

$(*)=\left\frac{e^{2x}}{2}\right]^3_1=\frac{e^{2(3)}}{2}-\frac{e^{2(1)}}{2}=\frac{e^6-e^2}{2}\ (units)$