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The sphere has a volume two-thirds that of the circumscribed cylinder and a surface area two-thirds that of the cylinder (including the bases). Since the values for the cylinder were already known, he obtained, for the first time, the corresponding values for the sphere. The volume of a sphere of radius r is 4 / 3 π r 3 = 2 / 3 ...
The ratio of the volume of a sphere to the volume of its circumscribed cylinder is 2:3, as was determined by Archimedes. The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder.
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form = (,,) where the are the coordinates, so that the volume of any set can be computed by = (,,).
Thin cylindrical shell with open ends, of radius r and mass m. This expression assumes that the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube for r 1 = r 2. Also, a point mass m at the end of a rod of length r has this same moment of inertia and the value r is called the radius of gyration.
Two common methods for finding the volume of a solid of revolution are the disc method and the shell method of integration.To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness δx, or a cylindrical shell of width δx; and then ...
The shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the xy-plane around the y-axis. Suppose the cross-section is defined by the graph of the positive function f(x) on the interval [a, b]. Then the formula for the volume will be: ()
In this case the volume of the band is the volume of the whole sphere, which matches the formula given above. An early study of this problem was written by 17th-century Japanese mathematician Seki Kōwa. According to Smith & Mikami (1914), Seki called this solid an arc-ring, or in Japanese kokan or kokwan. [1]
The cylindrical radius ρ of the point P is given by = + and its distances to the foci in the plane defined by φ is given by = (+) + = + The remaining coordinates μ and ν can be calculated from the equations cosh μ = d 1 + d 2 2 a cos ν = d 1 − d 2 2 a {\displaystyle {\begin{aligned}\cosh \mu &={\frac {d_{1}+d_{2}}{2a}}\\\cos \nu ...