A Sphere

(mathematics) A regular three-dimensional object in which every cross-section is a circle; the figure described by the revolution of a circle about its diameter [from 14th c.].
A spherical physical object; a globe or ball. [from 14th c.]
(astronomy, now rare) The apparent outer limit of space; the edge of the heavens, imagined as a hollow globe within which celestial bodies appear to be embedded. [from 14th c.]
(historical, astronomy, mythology) Any of the concentric hollow transparent globes formerly believed to rotate around the Earth, and which carried the heavenly bodies; there were originally believed to be eight, and later nine and ten; friction between them was thought to cause a harmonious sound (the music of the spheres). [from 14th c.]
(mythology) An area of activity for a planet; or by extension, an area of influence for a god, hero etc. [from 14th c.]
(figuratively) The region in which something or someone is active; one's province, domain. [from 17th c.]
(geometry) The set of all points in three-dimensional Euclidean space (or n-dimensional space, in topology) that are a fixed distance from a fixed point [from 20th c.].
(logic) The extension of a general conception, or the totality of the individuals or species to which it may be applied.

Rotate the upper semi-circle \(y = \sqrt{R^2 - x^2}\) from \(x = -R\) to \(x = R\) around the \(x\)-axis using the Disk Method:

\[V = \pi \int_{-R}^{R} \left(\sqrt{R^2 - x^2}\right)^2 dx = \pi \int_{-R}^{R} (R^2 - x^2) \, dx\]

By symmetry, evaluate over \([0, R]\) and double:

\[V = 2\pi \left[ R^2 x - \frac{x^3}{3} \right]_0^R = 2\pi \left( R^3 - \frac{R^3}{3} \right) = \frac{4}{3}\pi R^3\]