如何计算椭球体的体积？

测量是数学的一个子分支，用于讨论不同类型的几何形状，如立方体、圆柱体、椭圆体、它们的面积和体积。它用于找到用于测量对象或形状的各个方面的几何和代数方程，如表面积、体积、曲线表面积等。在测量中，体积用于计算 3D 对象内部的空间量可以填充。我们可以找到任何固体物体的体积，如正方形、圆柱体、椭圆体等。在本文中，我们将学习如何计算椭圆体的体积。

椭球体

椭球体是一个三维几何图形。它是椭圆形的封闭表面，被视为结构化椭圆。它的名字来源于椭圆，因为任何穿过椭圆体的平面都会形成一个椭圆。它具有三个旋转对称轴，这三个轴相互垂直并在一个点相交，该点称为椭圆体的中心。椭球有两种类型：

• 扁椭球：如果 a = b 且 a > c，则这种类型的椭球称为扁椭球。
• 长椭圆体：如果 a = b 且 c > a，则这种类型的椭球称为长椭圆体。

椭球的标准方程是

x²/a² + y²/b² + z²/c² = 1

这里 a ≠ b ≠ c。如果 a = b = c，则该椭圆体称为球体。

椭球体的体积

椭球体的体积是椭球体的量度，它表示由封闭曲面包围的三维空间的大小。

我们知道椭球方程是 (x ² /a ² ) + (y ² /b ² ) + (z ² /c ² ) = 1 其中a, b, c是椭球半轴的长度然后体积可以通过以下公式计算 -

Volume of Ellipsoid = (4/3) × π × a × b × c

扁椭球体的体积是

Volume of Oblate Ellipsoid = (4/3) × π × a × a × b

长椭圆体的体积是

Volume of Prolate Ellipsoid = (4/3) × π × a × b × b

Example:

Given the length of semi-axes are 5cm, 6cm, 4cm

So the volume of the ellipsoid is 

V = (4/3) × π × a × b × c

= (4/3) × π × 5 × 6 × 4

= 430/3

= 160

Hence the volume of the ellipsoid is 160

确定椭球体的体积

As we know that the equation of ellipsoid is 

(x²/a²) + (y²/b²) + (z²/c²) = 1

Let us assume that -a ≤ x ≤ a

Now, we cut the ellipsoid with a plane parallel to the yz-plane

So, we get an ellipse 

(y²/b²) + (z²/c²) = 1 – (x²/a²) 

(y²/b²(1 – (x²/a²))) + (z²/c²(1 – (x²/a²) )) = 1 

So the semiaxes are

p = b√(1 – (x²/a²)) and q = c√(1 – (x²/a²))

As we know that the area of ellipse is 

A(x) =  πbc(1 – (x²/a²)) …..(1)

Now by using the formula of parent entry we calculate the volume of the ellipsoid

V = \limi$\int^{a}_{-a}{A(x)} dx$

Now put the value of A(x) fro equation (1), we get

V = πbc \limi$\int^{a}_{-a}{(1-\frac{x^2}{a^2})} dx$

V = 4/3πbc

示例问题

问题1：如果半轴的长度是3cm、4cm、2cm，求椭球的体积。

解决方案：

Given,

Lengths of semi axes of an ellipsoid a=3cm, b=4cm, c=2cm

Volume = (4/3) × π × a × b × c

= (4/3) × π × 3 × 4 × 2

= 32 × π

= 100.53 cm³

So, volume of ellipsoid with given measurements is 100.53cm³.

问题2：如果半轴的长度是5cm、3cm、2cm，求椭球的体积。

解决方案：

Given,

Lengths of semi axes of an ellipsoid a = 5cm, b = 3cm, c = 2cm

Volume = (4/3) × π × a × b × c

= (4/3) × π × 5 × 3 × 2

= 40 × π

= 125.66 cm³

So, volume of ellipsoid with given measurements is 125.66cm³.

问题3：如果轴的长度是6cm、4cm、2cm，求椭球的体积。

解决方案：

Given,

Lengths of axes of an ellipsoid are 6cm, 4cm and 2cm.

Length of semi axes = Length of axes/2

a = (6/2) = 3cm

b = (4/2) = 2cm

c = (2/2) = 1cm

Volume = (4/3) × π × a × b × c

= (4/3) × π × 3 × 2 × 1

= 8× π

= 25.13 cm³

So, volume of ellipsoid with given measurements is 25.13cm³.

问题 4：如果轴的长度分别为 12cm、6cm 和 2cm，求椭球的体积。

解决方案：

Given,

Lengths of axes of an ellipsoid are 12cm, 6cm and 2cm.

Length of semi axes = Length of axes/2

a = (12/2) = 6cm

b = (6/2) = 3cm

c = (2/2) = 1cm

Volume = (4/3) × π × a × b × c

= (4/3) × π × 6 × 3 × 1

= 24× π

= 75.4 cm³

So, the volume of ellipsoid with given measurements is 75.4cm³.

问题 5：如果方程为 (x ² /7 ² ) + (y ² /4 ² ) + (z ² /2 ² ) = 1，求椭圆体的体积

解决方案：

Given,

Equation of ellipsoid, (x²/7²) + (y²/4²) + (z²/2²) = 1

It is of form (x²/a²) + (y²/b²) + (z²/c²) = 1

From this we can derive lengths of semi axes. 

a = 7

b = 4

c = 2

Volume = (4/3) × π × a × b × c

= (4/3) × π × 7 × 4 × 2

= (224/3) × π

= 234.57 cm³

So, the volume of ellipsoid with given measurements is 234.57cm³.