area of spherical zone
Let us consider the circle
(x-r)2+y2=r2 |
with radius r and centre (r, 0). A spherical zone may be thought to be formed when an arc of the circle rotates around the x-axis. For finding the are of the zone, we can use the formula
A= 2π∫bay√1+(dydx)2𝑑x | (1) |
of the entry area of surface of revolution. Let the ends of the arc correspond the values a and b of the abscissa such that b-a=h is the of the spherical zone. In the formula, we must use the solved form
y=(±)√rx-x2 |
of the equation of the circle. The formula then yields
A= 2π∫ba√rx-x2√1+(r-x√rx-x2)2𝑑x= 2π∫bar𝑑x= 2πr(b-a). |
Hence the area of a spherical zone (and also of a spherical calotte) is
A= 2πrh. | (2) |
From here one obtains as a special case h=2r the area of the whole sphere:
A= 4πr2. | (3) |
Remark. The formula (2) implies that the centre of mass of a half-sphere is at the halfway point of the axis of symmetry (h=r2).
Title | area of spherical zone |
---|---|
Canonical name | AreaOfSphericalZone |
Date of creation | 2013-03-22 18:19:05 |
Last modified on | 2013-03-22 18:19:05 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 9 |
Author | pahio (2872) |
Entry type | Derivation |
Classification | msc 26B15 |
Classification | msc 53A05 |
Classification | msc 51M04 |
Synonym | area of spherical calotte |
Related topic | AreaOfTheNSphere |
Related topic | CentreOfMassOfHalfDisc |