Question

The region bounded by $f\left(x\right)=4x-x^2$ and the x-axis between $x=1$ and $x=4$ is rotated about the y-axis. The volume of the solid generated is

• A. $\frac{81}{2}\pi$
• B. $\frac{41}{2}\pi$
• C. $\frac{83}{2}\pi$
• D. $12\pi$

Answer 1

The correct option is A $\frac{81}{2}\pi$

The required volume will be
$=2\pi {\int }_1^{4}xf\left(x\right).dx$
$=2\pi {\int }_1^{4}4x^2-x^3.dx$
$=2\pi {[\frac{4x^3}{3}-\frac{x^4}{4}]}_1^4$
$=2\pi [\frac{4^4}{3}-\frac{4^4}{4}-(\frac{4}{3}-\frac{1}{4}\left)\right]$
$=2\pi [\frac{4^4}{12}-\frac{13}{12}]$
$=\frac{\pi }{6}[4^4-13]$
$=\frac{\pi }{6}\left[243\right]$
$=\frac{81\pi }{2}$