Question

Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated.
Find the volume of the solid generated by revolving the region bounded by the curve y = ln x, the x-axis, and the vertical line x=e². about the x-axis. (Express the answer in exact form.)

Answer 1

Answer:

Step-by-step explanation:

Given that the region bounded by x axis, x=e^2 and y = lnx be rotated about x axis

The limits of x are $1$

Volume generated = $\int\limits^{e^2} _{1} \pi {y^2} \, dx \\\\=\int\limits^{e^2} _{1} \pi {(lnx)^2} \, dx \\\pi[x (lnx)^2 -\int\limits^{e^2} _{1} x(2lnx)dx]\\=2(e^1-1)\pi$