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 = 1, y = 5, x = 0; about the y-axis

Answer 1

Using the disk method, the volume is obtained with the integral

$\displaystyle\pi\int_{y=1}^{y=5}\left(\frac{e^y}3\right)^2\,\mathrm dy=\frac\pi9\int_1^5e^{2y}\,\mathrm dy$

The radius of each disk is given by the horizontal distance from the axis of revolution, $x=0$, to the logarithmic curve which we can write as a $x(y)$:

$y=\ln(3x)\implies e^y=e^{\ln3x}=3x\implies x=\dfrac{e^y}3$

Then the volume of one such disk is $\pi x(y)^2$.

The volume of the entire solid would be

$\displaystyle\frac\pi9\int_1^5e^{2y}\,\mathrm dy=\frac\pi{18}\int_1^52e^{2y}\,\mathrm dy=\frac\pi{18}\int_1^5e^{2y}\,\mathrm d(2y)$

$=\dfrac\pi{18}\left(e^{10}-e^2\right)$