Triple Integrals in Cartesian Coordinates

Solved Problems

Example 3.

Calculate the triple integral where the region (Figure ) is bounded by the surfaces

Solution.

The projection of the solid region onto the -plane looks as shown in Figure

Region U bounded by the surfaces z=xy, y=x, x=0, x=1, z=0 — Figure 5.

Region of integration in the xy-plane — Figure 6.

Taking this into account, we find the corresponding iterated integral:

Example 4.

Express the triple integral in terms of iterated integrals in six different ways. The region lies in the first octant and is bounded by the cylinder and the plane (Figure ). Find the value of the integral.

Solution.

Region U liying in the first octant and bounded by the cylinder x^2+z^2=4 and the plane y=3 — Figure 7.

Rectangular region of integration — Figure 8.

If the order of integration is then the iterated integral can be written as

Similarly, we can write the iterated integral to carry out the integration in the order

Now we consider the case i.e. when the first inner integral is taken over the variable Then

Since the projection of the solid onto the -plane is a rectangle (Figure ), then by changing the order of integration over and we have

Finally we can write the iterated integral in the order (starting from the inner integral):

The last th way looks as follows:

We can use any of the iterated integrals to calculate the value of the initial triple integral. Taking the last one, we get

Make the substitution:

As a result, we obtain:

It is easy to check that this value is just of the volume of the cylinder.