Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

Answer 1

Answer:

$\frac{\pi}{18}(e^{10} - e^8) \approx 3324$

Step-by-step explanation:

We begin by solving for x in term of y

$y = ln(3x)$

$e^y = 3x$

$x = \frac{e^y}{3}$

We can use the disk method to calculate the volume of rotation around the y axis

$V = \int\limits^5_4 {\pi r^2} \, dy$

Where $r = x = \frac{e^y}{3}$

$V = \int\limits^5_4 {\pi \left(\frac{e^y}{3}\right)^2} \, dy\\V = \frac{\pi}{9}\int\limits^5_4 {e^{2y}} \, dy\\\\V = \frac{\pi}{9}\left[\frac{e^{2y}}{2}\right]^5_4\\V = \frac{\pi}{9}(e^{10}/2 - e^8/2) = \frac{\pi}{18}(e^{10} - e^8)\\ V = \frac{\pi}{18}19045.5 \approx 3324$