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Triple Integral

Triple Integral

We are living in a three dimensional space. So we need to evaluate volumes.

Box (Source: clker.com)

This we can realize with triple integrals.

We know the integrals of areas. In this case the values  are summed up over little line segments .

If you sum up the values  over little areas  or , you take the double integral. The result is a volume.

At last the triple integral sums up three dimensional values  over little volumes  or .

Detailed explanations for this new sign:

We can define the integral of a voume with Riemann sums. We chop up a solid objekt into small boxes with the dimensions .

If this object is a cube, this chopping look like this.

Box consisting of small cubes

Formula of the volume of one of the small cubes.

The small volume  is the product of the small cubes . The sum of theses small cubes is the entire volume V of the box.

The big  leads to fine integral  by using the limes operation.

The entire volume of the box is an infinitesimal summation of the small cubes.

You treat the last integral as you do with the double integral. Evaluate first with respect to the most inner variable, while treating the other variables as constants.

As you now know what to do, the brackets are superfluous.

There are boundaries for each dimension.

a-b is the boundary for dx.

c-d is the boundary for dy.

e-f is the boundary for dz.

  Example

Evaluate the triple integral of  over the volume, which is determined by the Box B.

Integrate with respect to dx.

This is the result of integration:

Put in the boundaries of x and subtract.

Integrate with respect to dy.

This is the result of integration:

Put in the boundaries of y and subtract.

Integrate with respect to dz.

This is the result of integration:

After putting in the boundaries of z and subtraction: