The triple integral is a major calculation in integration, and if you are a student in pure and applied mathematics, you will have to handle it at one point or the other. Now lets us take a backward step to basic integration. We clearly understand integration as the opposite of differentiation. It is also extremely important in computing the area of a shape or region.
The triple integral calculation in mathematics gives students a comprehensive understanding of a shape or region’s area. It is an advancement from the single and double integration because it deals with three variables simultaneously. This means that students are handling up to three variables at a time.
Knowing how to compute Triple integral is essential if yu are in applied and pure mathematics because you will get to handle them a lot. In this piece, you will get to learn all the processes of computing the triple integral in the easiest way possible. You only need to pay attention to every step and continue practising with different examples.
That being said, here is all you need to know
What is a Triple Integral?
The Triple Integral is the limit of the product sum of a function multiplied by the volume of the rectangular solids.
For a more formal definition:
Let a three-variable continuous function given as f(x,y,z) be defined over T, a solid field. Then the triple integral over T is defined as
where the sum is taken over the rectangular solids included in T.
What is a Triple Integral Used For?
The Triple integral is used to ensure a better understanding of multiple fields. It is, however, very important to have a very excellent understanding of integration to calculate the area of the interested regions.
Several important considerations come with the triple integral, and we will outline them below.
Property 1: The Triple Integral is simply an additional integration to the normal double integration
Property 2: The Triple integral is always closed with a lower and upper limit. As such, its final answer will always be a real number
Property 3: The integration is generally systematic. As such, the first integration will be dependent on the first differentiable variable. The second will be dependent on the outlined differentiable variable, and the third integration will be dependent on it being the third differentiable variable.
Now we will attempt to explain this below
As we can see, the first variable to be integrated will be x because dx is before dy and dz. In accordance, y will automatically be the second variable to be integrated because dy is before dz, and finally, the last integration will be in respect to z because dz comes last.
So how do you calculate the triple integration? The next heading will outline the respective steps.
How Do You Do Triple Integral Problem
To calculate the triple integral easily, below are the steps to follow.
Step 1: Determine the first variable to integrate by checking the order of the dx, dy and dz. It is not compulsory that dx will always be the first differentiation symbol in a triple integral problem. As such, it is essential that you pay attention to the arrangement.
Step 2: When integrating with respect to a particular variable, all other variables will be treated as constants. For example, if you are integrating in respect to x, the variables y and z will be treated as constants.
Step 3: you must conclude a variable integration by implementing its limit requirement. This will help eliminate that variable and leave the ones that are yet to be integrated. This rule must be followed until the last integration. This is why the triple integration will always end with a real number as its final answer.
It is important to note that all the normal integration rules also stand for triple integration.
How to Calculate the Triple Integral with Examples?
While we will consider several examples of Triple Integral, the first two examples will consider the single integration when it is bounded.
Calculate the triple integral
and ensure that your answer is expressed as a real number or real numbers.
So we will outline the steps as we solve this example.
Step 1: Determine the first variable to integrate by checking the order of the dx, dy and dz.
Based on the first step, we will integrate ex because dx comes first before any differentiable variable
Step 2: When integrating in respect to a particular variable, all other variables will be treated as constants. For example, if you are integrating in respect to x, the variables y and z will be treated as constants.
Now that we have the integration for x already, we will consider the last step outlined
Step 3: you must conclude a variable integration by implementing its limit requirement. This will help eliminate that variable and leave the ones yet to be integrated.
Since we have the bounds for x variables as 0 and 2, we will have
Since we are done with the x variable, we will move to the next two variables with our new solutions.
So we now have
With x variable totally non-existent in the new integral, we will be moving to the second bounded limits of -1 and 2, which is actually focused on the y variable. That will leave us with a repetition of the process we used to integrate the x.
So we now have
Now we have the last z variable to integrate with bounded limits given as 2 and 3. So we have
Clearly from the above, we have finally eliminated both the x and y variables, leaving us with the z variable. This means we will also use the farthest integral, which we can consider as the last integral.
As such, we have
We have now finally arrived at a real number solution after completing the triple integration. This confirms the fact that the triple integration or triple integral will generally lead to a real number solution.
As a formal conclusion, we can say that;
This is the final answer for this particular example.
We will consider another example to confirm the method of calculating the triple integral.
Calculate the Triple Integral and arrive at a real number solution
From the problem, we can see that the variable Z will need to be considered first before the variable y and finally the variable x. So we integrate each variable while keeping the others as constants in the same order.
For the Z variable, we will have
We will use the integral with bounded limits of 2 and 3 (the first integral closest to the variables) to get the real number of our Z variable.
Now, for the y variable, we have
To solve the above integration, we have to use the U substitution, a quite popular Integration technique.
Since it is impossible to integrate the x variable because it is hooked under an order of 2 with other constant, the U substitution will be employed.
Now let U = 1 + 2x.
That automatically changes our integration to
Now we can’t make any form of integration because the U variable does not match the dx requirement. We will need to leave the integral for now and move forward. We will come back to it in a moment
Now we will have to take the differentiation of U = 1 + 2X
du/dx = 0 + 2
du/dx = 2
du/2 = dx …. (i)
now remember our integration given as
Now remember u = 1 + 2x
That makes our above-integrated solution
Are Triple Integral 4D?
Actually yes. Triple integral help to provide information on shapes with 4D coordinates. The double integral provides information on 3d coordinates, and the single basic integral is 2D.
Computing triple integrals are not as complicated as seen throughout this article. However, you need to know exactly what is required. This is exactly why we take you through a step-by-step calculation to build your confidence and feel more comfortable with its philosophy. Generally, the triple integral is not exactly different from normal integration computations.
The only difference is that; you will need to perform integration on three variables to get your solution. However, as long as you are careful, there will be no issues or complications. You can continue to use the knowledge gained here to solve other examples and problems.