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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}}

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