Problem Statement
The region enclosed by the curve $y = \ln x$, the $y$-axis, and the horizontal lines $y = 0$ and $y = 1$ is rotated through $2\pi$ radians about the $y$-axis.
Find the exact volume of the solid generated. [6 Marks]
Verified Solution & Marking Scheme
Express x in terms of y
$$y = \ln x \implies x = e^y$$
Set Up Volume of Revolution Integral about y-axis
$$V = \pi \int_c^d x^2 \, dy = \pi \int_0^1 (e^y)^2 \, dy = \pi \int_0^1 e^{2y} \, dy$$
Evaluate Definite Integral
$$V = \pi \left[ \frac{1}{2} e^{2y} \right]_0^1 = \frac{\pi}{2} \left( e^{2(1)} - e^0 \right) = \frac{\pi}{2}(e^2 - 1)$$