Lecture №8

# Lecture №8. The triple integral and its properties. The geometric meaning. The calculation of triple integrals

Exercises for Lecture #8

The purpose of work:

The acquaintance with the triple integral and its properties.

Theoretical information and examples

Content

2.7 Triple Integrals

2.7.1 Triple integrals in rectangular co-ordinates

2.7 Triple Integrals

2.7.1 Triple integrals in rectangular co-ordinates. The triple integral of the function f(x, y, z), extended over the region V, is the limit of the corresponding three-fold iterated sum

Evaluation of a triple integral reduces to the successive computation of the three ordinary (simple iterated) integrals or to the computation of one double and one single integral.

Example 1. Compute

,

where the region V is defined by the inequalities

Solution.

Example 2. Evaluate

extended over the volume of the ellipsoid

Solution.

where Syz is the area of the ellipse

which equals

Hence, we finally get

.