Problem Statement
The curve $y = \ln x$ for $1 \le x \le e^4$ is rotated through $2\pi$ radians about the $y$-axis to form a solid bowl.
**(a)** [2 marks] Express $x$ in terms of $y$ and state the bounds for $y$.
**(b)** [5 marks] Calculate the exact volume of the solid formed in terms of $\pi$ and $e$.
Verified Solution & Marking Scheme
Part (a) [M1]
$y = \ln x \implies x = e^y$
Part (a) [A1]
When $x = 1$, $y = \ln 1 = 0$. When $x = e^4$, $y = \ln(e^4) = 4$. Bounds: $0 \le y \le 4$
Part (b) [M1]
Volume formula about $y$-axis: $V = \pi \int_{c}^{d} x^2 \, dy$
Part (b) [A1]
$V = \pi \int_{0}^{4} (e^y)^2 \, dy = \pi \int_{0}^{4} e^{2y} \, dy$
Part (b) [M1]
$\pi \left[\frac{e^{2y}}{2}\right]_{0}^{4}$
Part (b) [A1]
$V = \frac{\pi}{2}(e^8 - 1) \text{ cubic units}$