# Calc5.1

Find the volume of the solid generated by revolving the line $y=x$ around the x-axis, where $0\leq x\leq 4$.
This is a cone with radius 4 and height of 4. Since the volume of a cone is given by the formula $V={\frac {1}{3}}\pi r^{2}h$, we should get a volume of ${\frac {64\pi }{3}}$. Our axis of revolution is the x-axis so our radius function is a function of x. Moreover, it is $r(x)=x$ since the distance from our axis of revolution to the function is always x. Thus, the volume is given by
$V=\pi \int _{{0}}^{{4}}x^{2}\,dx=\pi \left[{\frac {1}{3}}x^{3}\right]_{{0}}^{{4}}={\frac {64\pi }{3}}$