Volume of a Three Dimensional Region in Space
The volume of a closed bounded region D in space is the value of the integral
Example:
Find the volume of the three-dimensional region enclosed by the surfaces
and
Solution:
Average Value of Function in Space:
The average value of function F over a region R in space is defined by
the formula:
Average value of F over R
Example:
Find the average value of F (x, y, z) = xyz over the cube bounded by the coordinate planes and the planes x = 2, y = 2, z = 2 in first octant.
Solution:
The volume of the cube is (2)(2)(2) = 8 and
The value of the integral of F over the cube is:
Average value of xyz over the cube
Exercise:
Find the following integrals:
(1)
(2)
(3)
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