Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = ln 3x, y = 4,

Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = ln 3x, y = 4, y = 5, x = 0; about the y-axis
The answer I got was (pi/18)(e^16-e^8) and that is wrong.

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1 week 2022-01-07T06:34:31+00:00 1 Answer 0 views 0

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    2022-01-07T06:35:43+00:00

    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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45:7+7-4:2-5:5*4+35:2 =? ( )