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Volume of revolution

When a graph is rotated about the $x$ -axis, it forms a solid of thin circular discs. Each disc has radius equal to the function value, so the area of the disc is $\pi$ times the radius squared. The volume is the integral of that area. For $y = \sqrt{x}$ the radius squared equals $x$ , and the integral is simple. The cone formula is the same idea with a straight line as the radius.

V = \pi\int_a^b f(x)^2\,dx

volume of revolution about the x-axis

Symbols

$f(x)$	the radius of the disc
$V$	the volume

Example

$y = \sqrt{x}$ from $0$ to $4$ gives $V = \pi\int_0^4 x\,dx = \pi\cdot\dfrac{4^2}{2} = \pi\cdot 8 \approx 25.13$ .

It is the function value, not the integral, that is squared inside the integral.

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Part of Mathematics R2: Area and volume with integrals.