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AURORKA [14]
3 years ago
5

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

Mathematics
1 answer:
Elza [17]3 years ago
4 0

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

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