# How do you find the volume bounded by y = 12 ln x, the x-axis, the y-axis and the line y=12 ln14 revolved about the y-axis?

Jun 21, 2016

$1170 \pi$

#### Explanation:

Working up the y axis, the elemental volume of a small disc of thickness $\Delta y$ revolved about the y axis will be $\Delta V = \setminus \pi {x}^{2} \Delta y$

where: $x = {e}^{\frac{y}{12}}$ because $y = 12 \ln \left(x\right)$ ..... and so ${x}^{2} = {e}^{\frac{y}{6}}$

so $V = \pi \setminus {\int}_{0}^{12 \ln \left(14\right)} \setminus {e}^{\frac{y}{6}} \setminus \mathrm{dy}$
$= 6 \setminus \pi {\left[{e}^{\frac{y}{6}}\right]}_{0}^{12 \ln \left(14\right)}$
$= 6 \pi \left[\exp \left(\frac{12 \ln \left(14\right)}{6}\right) - 1\right]$
$= 6 \pi \left[{e}^{2 \ln \left(14\right)} - 1\right]$
$= 6 \pi \left[{e}^{\ln \left({14}^{2}\right)} - 1\right]$
$= 6 \pi \left[{14}^{2} - 1\right] = 1170 \pi$