1. Triple Integral

2. 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.

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

4. 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.

5. 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

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

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

8. 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:

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

10. 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

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

12. Volume

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

15. Example-3

17. Class work