Triple Integrals

By Author: math qa22
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The symbol
dx dy dz denotes a triple integral.
If x1 and x2 are constants, y1 and y2 are either constants or functions of x and z1 and z2 are either constants or functions of x and y, then this integral is evaluated as follows:

where the integration is carried out from the innermost rectangle to the outermost one.

Thus, first f(x,y,z) is integrated with respect to z between the limits z1 and z2 keeping x and y fixed. The resulting expression is integrated with respect to y between the limits y1 and y2 keeping x constant. The result just obtained is finally integrated with respect to x from x1 to x2.

The order of integration may be different for different types of limits.

Examples of triple integrals

1. Evaluate the following triple integral:

Solution :

Application of triple integrals.
Volume as triple integrals

Divide a solid into rectangular parallelopiped of volume by planes parallel to the co-ordinate planes. In order to find the volume of the solid, first take the sum of the elementary parallelepipeds in a vertical column PQ parallel to z ...
... axis. The result is:
𝜕x𝜕y , where z1 and z2 being the values of z at the ends of the column.

Next sum these columns along the base MN, parallel to the y axis and thus obtain the value of the plane MNR.

The result is
𝜕x = 𝜕x
Where y1 and y2 are the limiting values of y in the plate.
Finally take the sum of these plates, parallel to the z axis.
The result is:
dx dy dz
The limit of this sum is x,y,z. V will be the volume of this solid in the form
V =
The differential of the volume in the rectangular co-ordinates is
dV = dx dy dz
which can be used in finding the moment of inertia, centre of gravity etc. of solids.

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