Question

Describe the method of establishing and evaluating a triple integral. Give an example using ONE of the following options: (a) iterated triple integral, (b) use of spherical coordinates, (c) use of cylindrical coordinates.

Answer 1

A triple integral dV is evaluated over a closed, bounded solid region D in . Suppose f(x,y,z) is defined on a closed region D, which in turn is contained in a box. Divide the box into finite number of smaller units and choose a representative point (x',y',z') from each of the box.

From the Riemann sum that is .

As the norm approaches 0, we get :

which is our triple integral.

The most basic example of Iterated Integral is one where the box B is nothing but a rectangular solid. Where . The integral will be :

.