Triple Integral

TRIPLE INTEGRALS Just as we defined -single integrals for functions of one variable double integrals for functions of two variables, so we can define triple integrals for functions of three variables.

Consider f (x,y,z) defined on a rectangular box:

The first step is to divide B into sub-boxes—by dividing: The interval [a, b] into l subintervals [xi-1, xi] of equal width Δx. [c, d] into m subintervals of width Δy. [r, s] into n subintervals of width Δz.

The planes through the endpoints of these subintervals parallel to the coordinate planes divide the box B into lmn sub-boxes -Each sub-box has volume ΔV = Δx Δy Δz

Then, we form the triple Riemann sum where the sample point is in Bijk.

The triple integral of f over the box B is: if this limit exists. Again, the triple integral always exists if f is continuous.

We can choose the sample point to be any point in the sub-box. However, if we choose it to be the point (xi, yj, zk) we get a simpler-looking expression:

FUBINI’S Theorem (TRIPLE INTEGRALS) If f is continuous on the rectangular box B = [a, b] x [c, d] x [r, s], then

The iterated integral on the right side of Fubini’s Theorem means that we integrate in the following order: With respect to x (keeping y and z fixed) With respect to y (keeping z fixed) With respect to z

Evaluate the triple integral Example 1 Evaluate the triple integral where B is the rectangular box

Volume

Example 2 Find the volume of a cube of length 4 units using triple integral.

Example-3

Class work