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by：XY Screens
2021-08-26

The definition of the sphere definition: The figure composed of all points whose distance to a fixed point in space is equal to the fixed length is called a sphere. The spherical surface is a three-dimensional figure with a continuous curved surface, and the geometric body enclosed by the spherical surface is called a sphere. The composition of the sphere The surface of the sphere is a curved surface, and this curved surface is called a spherical surface. The sphere is similar to a circle, and it also has a center called the center of the sphere. Astral, especially the 'earth'. The sphere in mathematics The basic concept of sphere The diameter of a semicircle is its axis of rotation, and the curved surface formed by rotation is called a spherical surface. The geometric body enclosed by the sphere is called a sphere, or sphere for short. The center of the semicircle is called the center of the sphere. The line segment connecting the center of the sphere and any point on the sphere is called the radius of the sphere. The line segment connecting two points on the sphere and passing through the center of the sphere is called the diameter of the sphere. Spherical properties Using a plane to cut a sphere, the cross section is a round surface. The cross-section of the ball has the following properties: 1 The line connecting the center of the sphere and the center of the cross-section is perpendicular to the cross-section. 2 The distance d from the center of the sphere to the cross-section has the following relationship with the radius R of the sphere and the radius r of the cross-section: r^2u003dR^2-d^2 The circle intercepted by the plane passing through the center of the sphere is called a great circle. The circle cut through the cross section of the center of the sphere is called a small circle. On the sphere, the length of the shortest line between two points is the length of a minor arc between the two points of the great circle passing through these two points. We call this arc length the spherical distance between the two points. Sphere function The function of a sphere with a radius of r is: r^2u003dx^2+y^2+z^2 The formula for calculating the volume of a sphere with a radius of R is: Vu003d(4/3)πR^ 3 (four-thirds times π times the cube of the radius) Vu003d(1/6)πd^3 (one-sixth times π times the third power of the diameter) The formula for calculating the surface area of u200bu200ba sphere with a radius of R Yes: Su003d4πR^2 (4 times π times the square of R) Sphere calculation proof: Proof: V ballu003d4/3*pi*r^3 Want to prove V ballu003d4/3pi*r^3 , It can be proved that V hemisphereu003d2/3pi*r^3 make a hemisphere hu003dr, make a cylinder hu003dr ∵V column-V cone u003d pi*r^3- pi*r^3/3 u003d2/3pi *r^3 ∴If the conjecture is true, then V column-V cone u003d V hemisphere∵According to Cavalieri principle, two three-dimensional figures sandwiched between two parallel planes are parallel to any plane of these two planes The cut, if the two cross-sectional areas obtained are equal, then the volumes of the two three-dimensional figures are equal. ∴If the conjecture holds, two planes: S1(circle)u003dS2(ring) 1. Cut a plane from the hemispherical height h point. According to the formula, the area is pi*(r^2-h^2)^0.5^2u003d pi*(r^2-h^2) 2. Make a cone with the same base and the same height from the cylinder: According to the formula, the area of u200bu200bthe ring on the right side of the V cone is pi*r^2-pi*r*h/r u003dpi*(r^2-h^2) ∵pi*(r^2-h^2)u003dpi*(r^2-h^2) ∴V column-V coneu003dV hemisphere∵V column-V Coneu003dpi*r^3-pi*r^3/3u003d2/3pi*r^3 ∴V hemisphereu003d2/3pi*r^3 can be derived from the V hemisphere V sphereu003d2*V hemisphereu003d4/3 *pi*r^3 Of course, there are many ways to find the volume of a sphere. It is easier to understand the solution by reintegration: the integration area is calculated as the sphere as shown in the figure, and the radius of the circle is r

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