Volume of Cone Using Volumes of Revolution

The formula for the volume of a cone can be derived using volumes of revolution. For a cone of base radius  
\[r\]
  and height  
\[h\]
  we can rotate the line  
\[y= \frac{r}{h} x, \; 0 \le x \le h\]
  about the  
\[x\]
  axis.

volume of cone using volumes of revolution

The volume of revolution forms a cone. The volume of the cone is
\[\begin{equation} \begin{aligned} V &= \pi \int^h_0 y^2 dx \\ &= \pi \int^h_0 (\frac{r}{h} x)^2 dx \\ &= \pi \frac{r^2}{h^2} \int^h_0 x^2 dx \\ &= \pi \frac{r^2}{h^2} [\frac{x^3}{3} ]^h_0 \\ &=\frac{1}{3} \pi r^2 h \end{aligned} \end{equation}\]

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