Show that $\frac{\mathrm{d}}{\mathrm{d}x}(x \ln x - x) = \ln x$.[2] A curve $C$ is given by

Question

Show that $\frac{\mathrm{d}}{\mathrm{d}x}(x \ln x - x) = \ln x$.[2] A curve $C$ is given by the equation $y = \ln x$. The region $R$ is bounded by $C$, the $x$-axis and the line $x = \mathrm{e}$. Using the result in part (a), find the exact volume generated when $R$ is rotated $360^\circ$ about the $x$-axis. [4] Find the volume generated when $R$ is rotated $360^\circ$ about the $y$-axis, giving your answer correct to $2$ decimal places.[3]

Accepted Answer

(b) $\pi(\mathrm e-2)$ (c) $\dfrac{\pi}{2}(\mathrm e^2+1)\approx13.18$

N2025 P1 Q7

N2025 P1 Q7

Video Solution:

Solution 1