Science:Math Exam Resources/Courses/MATH103/April 2016/Question 05 (a)/Solution 1

The formula for the volume of the solid obtained by revolving $R$ around the y-axis is

$V=\int _{a}^{b}\pi f(y)^{2}\,dy,$

where $f(y)$ is the curve defined by the graph of $y=-\ln(x),$ expressed as a function of $y.$

Solving for $x$ in $y=-\ln(x)$ yields $x=e^{-y}$, i.e., $f(y)=e^{-y}.$

The graph of the region $R$ also shows that $y$ varies between $0$ and $1,$ so $a=0$ and $b=1.$ Hence by the formula

${\begin{aligned}V&=\int _{0}^{1}\pi e^{-2y}\,dy\\&=\left.\pi \cdot {\frac {e^{-2y}}{-2}}\right|_{0}^{1}\\&=-{\frac {\pi }{2}}\left(e^{-2}-1\right)\\&=\color {blue}{\frac {\pi }{2}}(1-e^{-2}).\end{aligned}}$