Lecture ¹9

# Lecture ¹9. The change of the variable in the triple integrals. Practical application of triple integrals Exercises for Lecture 9

The purpose of work:

The change of the variable in the triple integrals. Practical application of triple integrals.

Theoretical information and examples

Content

2.7.2 Change of variables in a triple integral

2.7.3 Applications of triple Integrals

2.7.2 Change of variables in a triple integral. If one has to change in the triple integral

from the variables x, y, z to the variables u, v, w, when

where the functions j, y, c are

1)continuous together with their first partial derivatives;

2) in (1,1) (and, in both directions, continuous) correspondence between the points of the region of integration V of xyz-space and the points of some region V' of UVW-space;

3) the functional determinant (Jacobian) of these functions

retains a constant sign in the region V, then we can use the formula

.

In particular,

1) for cylindrical co-ordinates (Fig.13), where

we have

2) for spherical co-ordinates (Fig. 14), j, y, r (j is the longitude, y the latitude, r the radius vector, Fig.15, where

we have .

Example 3. Change to spherical co-ordinates to compute

where V is a sphere of radius R.

Solution. For a sphere, the ranges of the spherical co-ordinates j (longitude), y (latitude) and r (radius vector) will be

whence we have

2.7.3 Applications of triple Integrals. The volume of a region of xyz-space is

.

The mass of a solid occupying the region V is

where g (x, y, z) is the density of the body at the point (x, y, z).

The static moments of a body relative to the co-ordinate planes are

The co-ordinates of the centre of gravity are

If the solid is homogeneous, we can set g(x, y, z) = l in the formulae for the co-ordinates of the centre of gravity.

The moments of inertia relative to the co-ordinate axes are

Setting g(x, y, z) = l in these formulae, we get the geometric moments of inertia of the body.