Changing the order of integration in a triple integral

 Goal: Change the order of integration in a triple integral over a  general bounded region.

Considerations

Changing the order of integration in double integrals facilitates their computations. In triple integrals over a rectangular box, changing the order of integration doesn't simplify the calculations. However, changing the order of integration in a triple integral over a general bounded region facilitates the computation.

Example

Solution

Let's start by defining the region E and the form in which the triple integral should be expressed once the order of integration is changed:

Let's draw the projections on the three coordinates plane:

Let's redefine E for the change of the order of integration:

The integral becomes:

Let's calculate the given integral by substituting f(x,y,z) by xyz.

= 1/168

Let's calculate the second integral with the order of integration changed.

The answers for both integrals are identical.

Finding a volume by evaluating triple integrals

 Goal: Find the volume of a solid using a triple integral.

Just as we used a double integral to find an area, we are going to use a triple integral to find a volume.

In the case of an area, we used the formula:

In the case of the volume, we are going to use:

Let's do an example:

Example

Solution

The volume of the pyramid is then 4/3 cubic feet.

Triple integrals over a bounded region

 We expand the definition of the triple integrals to calculate the triple integral of a function f(x,y,z) over a general bounded region E in R³. We start first by projecting this region on the xy plane.

Let D be the bounded region that is a projection of E. We define E as follows:

The  2 functions are shown in the figure above.

The following theorem defines the triple integral of the function f(x, y, z) over the region E.

Theorem

The triple integral of a continuous function f(x,y,z) over a general three-dimensional region

in R³, where D is the projection of E onto the xy-plane is

The calculation of the triple integral depends on which plane is the region D. is the region. In the theorem above the calculation is based on the region D located in the xy plane.. Now we consider the region D in the xz plane and two functions  y = u₁(x,y) and y = u₂(x.y) such that u₁(x,y) ≤ u₂(x,y) for all (x, z) in D. The region E can be defined as:

The region D in any plane can be of type I or type II as mentioned in double integrals over general

 regions. If D is of type I (figure below). Then the region E is defined as follows :                           :

                    

 

If D in the xy plane is of type II as shown in the figure below then the region E is define as follows:

Example                                                                                                                                                      

Solution

 

The following figures show the solid tetrahedron E and its projection on the xy plane. 

   

                       

 

             

This result is obtained by considering 5x-3y as a constant and then calculating  the integral                .

                 

The above result is obtained by differentiating with respect to y and factoring  .We may not factorize and obtain the same result                                                                                                                     

                                                                                                                

Notions of Triple Integrals

 Objectives

1) Recognize when a triple integral is integrable over a rectangular box

2) Evaluate a triple integral using an iterated integral

Integrable  functions of three variables

We are going to follow the same procedure as we did in double integrals.  We divide the interval [a,b} into l

the following figure:

As we partition the box into into smaller and smaller boxes, the number of small boxes becomes infinitely large. In this case the triple sum tends to a number called the triple integral of the function f(x,y,z).

Definition

Theorem

Now that we define the triple integral, the Fubini's theorem allows to evaluate it the same way it was used in the case of double integral.

Example I

Solution

The order of integration is known. Let's integrate with respect to x first then y and z.

Example II

Solution

Now let's integrate in a different order to see if we get the same answer. Let's integrate with respect to x, then z and y.

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Practice