Кратные интегралы. Учебное пособие

Fig. 2.1

Transferring to the repeated integral we get the final formula for calculation of a triple integral

Remarks.

1.If the region V is complex, then it should be splitted by finite number of correct regions to which the formula (2.2) is applied.

2.The order of integrating in the formula (2.2) can be another.

V is bounded by the surfaces = 0, = 0, = 0, + = 2, = 5. Solution. We construct given surfaces in the Oxyz coordinate space (fig.

2.2).

Fig. 2.2

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For the calculation of a triple integral so called cylindrical coordinates are often applied.

The position of a point M(x;y;z) in space Oxyz can be determined by three numbers r, φ, z where r is the length of a radius-vector projection onto Ox axis, z – applicata of a point M (fig. 2.3). These three numbers ( ; ; ) are called the cylindrical coordinates of a point M.

The cartesian and the cylindrical coordinates are connected by the

Fig.2.3

The right parts of equations (2.4) are continuously differentiable functions. The Jacobian of the coordinate’s transformation has the view

So, the formula (2.3) has the view

So, the calculation of the triple integral in cylindrical coordinates is led to the integration with respect to r, to φ and to z like in cartesian coordinates.

Remark. The cylindrical coordinates are convenient in the cases when the region of integrating is formed by the cylindric surface.

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V the region V is bounded by the surfaces = 1, 2 + 2 = 4, = 2 + 2 +2.

Solution. We construct given surfaces in the Oxyz coordinate space (fig.

2.4).

Fig. 2.4

From the fig. 2.4 clear that a given integral is convenient to calculate in the cylindrical coordinates expressed by the formulas (2.4).

The spherical coordinates of a point M(x;y;z) in space Oxyz are called the three numbers , , where is the length of a radius-vector of a point M, is an angle between the radius-vector projection onto Oxy plane and Ox

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f (x; y; z)dV =

Remark. The spherical coordinates are convenient in the cases when the region of integrating is sphere or its part. An equation for the sphere has the

x2 + y2 + z2 = 4 and the plane = 0 ( ≥ 0).

Solution. We find a given integral in the spherical coordinates using the formulas (2.6).

2.3.The Applications of the Triple Integral

Let’s consider some examples of the triple integral applications.

The Body Volume

The volume of the region V in the cartesian coordinates is expressed by the formula

V

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The Mass of a Body

If the volume density is given then the body mass is found by the triple integral

where ( ; ; ) is the volumetric density of mass distribution at a point

M(x;y;z).

The Static Moments and the Center of Mass Coordinates

of a Body

The static moments of a body relatively to Oxy, Oxz and Oyz coordinate planes respectively can be calculated by the formulas

Sxy = z (x; y; z)dxdydz,

V

Syz = x (x; y; z)dxdydz.

V

The static moments and the center of mass coordinates of a body are connected by equations

The moments of inertia relatively to Oxy, Oxz and Oyz coordinate planes can be found by the formulas

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Example 2.4. Find the volume, the mass, the static moment relatively to Oyz plane, the center of mass abscissa of a body bounded by the surfaces

= 2 + 2, = 4 if the density is ( ; ; ) = .

Solution. We construct given surfaces in the Oxyz coordinate space (fig.

2.6).

Fig. 2.6

We find the volume of a body by the formulas (2.8) and (2.9):

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