Volume of Circular Paraboloid

Theorem

Let $\CC$ be the region of space between:

a circular paraboloid $\PP$
a plane $\SS$ intersecting $\PP$ perpendicular to the axis $\AA$ of the parabola generating $\PP$.

Let $b$ be the radius of the circle which is the intersection of $\PP$ and $\SS$.

Let $a$ be the length of the line segment of $\AA$ between $\SS$ and the vertex of the parabola generating $\PP$.


The volume $\VV$ of $\CC$ is given by:

$\VV = \dfrac {\pi b^2 a} 2$


Proof


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Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Paraboloid of Revolution: $4.48$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Paraboloid of Revolution: $7.48.$
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