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Derive the formula for the volume of a right circular cone with radius $'r'$ and height $'h'$.

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  • Area bounded by the curve $t=f(x),$ the x-axis and the ordinates $x=a,x=b$ is $\int \limits_a^b f(x) dx $ or $ \int \limits _a^b y dx $
  • If the curve lies below the x-axis for $a \leq x \leq b,$ then the area is $\int \limits_a^b (-y) dx=\int \limits_a^b (-f(x))dx$
Step 1:
A right circular cone of radius r,height h, generated when the area bounded by the line OA(Where a is the point(h,r))
Step 2:
The x-axis and $x=h$ is rotated about the x-axis
Equation of OA is $\large\frac{y}{x}=\frac{r}{h}$$=>y=\large\frac{r}{h}$$x$
The volume of the cone$=\pi\int\limits_0^h y^2dx=\pi\int\limits_0^h \large\frac{r^2}{h^2}$$x^2dx$
answered Aug 16, 2013 by meena.p

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