Answer
$\frac{153\pi}{5}$
Work Step by Step
Setup the integration using disk method about the y-axis
$\pi \int_1^4 (-y^2 +4y)^2dy$
$\pi \int_1^4 (y^4 -8y^3 +16y^2)dy$, Integrate
$\pi [\frac{1}{5}y^5 -2y^4 + \frac{16}{3}y^3]_1^4$, Evaluate the definite integral
$\pi [(\frac{1024}{5} -512 + \frac{1024}{3}) - (\frac{1}{5} -2 +\frac{16}{3})]$, Simplify, or use your calculator to get a decimal approximation
$\approx 96.084$
or
$\frac{153\pi}{5}$
