Find the volume of a solid generated by revolving the region bounded by the given lines adn curves about the x-axis.
$\displaystyle y=6x,y=6,y=0$
So, you have $\displaystyle {y_1} = 6x,{\text{ }}{y_2} = 6,{\text{ }}x = 0,{\text{ }}{V_x} = ?$
Find the intersection point(s) of curves
$\displaystyle {y_1} = {y_2} \Leftrightarrow 6x = 6 \Leftrightarrow x = 1.$
Then you have
$\displaystyle {V_x} = \pi \int\limits_{x = a}^{x = b} {\left( {y_2^2 - y_1^2} \right)dx} = \pi \int\limits_0^1 {\left( {36 - 36{x^2}} \right)dx} = 36\pi \int\limits_0^1 {\left( {1 - {x^2}} \right)dx} =$
$\displaystyle = 36\pi \left. {\left( {x - \frac{{{x^3}}}{3}} \right)} \right|_0^1 = 36\pi \left( {1 - \frac{1}{3}} \right) = 24\pi {\text{ }}\left( {{\text{cubic units}}} \right).$
See this picture