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Find the volume of the figure enclosed by

$\begin{cases} y = \ln 5x \\ y=3\\ y=4\\ x=0\\ \end{cases}$

and rotated about the $y$-axis.

I came up with the following integral in terms of $y$:

$$\int_3^4 \pi \left(\dfrac{e^y}{5} \right) ^2dy$$

but the approximate answer of $161.95$ is incorrect. I asked a tutor at my university and they came up with the same answer.

Thanks in advance.

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    The answer is supposed to be $$\frac{\pi}{50}(e^{8}-e^{6}).$$ – 2017-01-19
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    Assuming you've stated the problem correctly, you and your tutor are right and the answer key is wrong. (Or maybe they want more digits or a symbolic answer if it's a computer submission?) – 2017-01-19
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    @vaponteblizzard I seem to get the answer you have provided too. However, would you mind stating what the answer key says so that we can see why it may be the case? – 2017-01-19
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    It was indeed a computer submission so I don't have an answer key. I finally was marked right after simplifying some terms in the exact answer. Usually, simplification is not necessary which is why I tried decimal form when it didn't work the first time. I was seriously starting to doubt myself! – 2017-01-20

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You are correct. I went ahead and found the volume using the cylinder method and I arrived at the same answer.

Note that $y=3$ intersects $\ln 5x$ at $(4.017, \ 3)$ and $y=4$ intersects it at $(10.92, \ 4)$, which you can see here on Desmos. Using this, we have:

$$ \displaystyle{2 \pi \int_0^{10.92} (4x) \ dx - 2\pi \int_{4.017}^{10.92} (x \ln 5x ) \ dx - 2\pi \int_0^{4.017} (3x) \ dx } $$ $$=161.951$$

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    It's refreshing to see it confirmed with the shell method! As I said in my comment above, it was a particularly persnickety computer submission. Thanks for the help. – 2017-01-20