Answer
$\pi(\frac{327}{4} - 30 \ln{4})$
Work Step by Step
Setup the integration using Disk Method with the region bounded by $xy=3, y=1, y=4, x=5 $
Put the equations in terms of y because of the rotation about the y-axis
$ \pi \int^4_1 (5-\frac{3}{y} ) ^2 dy$
$ \pi \int^4_1 (25- \frac{30}{y} + \frac{9}{y^2})dy$
$ \pi [25y - 30ln|y| - \frac{9}{y}] ^4_1 $ Evaluate the definite integral
$ \pi [(100 - 30\ln(4) -\frac{9}{4}) - (25-9)]$
$\pi(\frac{327}{4} - 30 \ln{4})$ Simplify
