Calculus - Volume of Solid of Revolutions

Question

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.

Answer 1

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:

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