# Triple Integral Calculator

The Triple Integral Calculator is a free online calculator that displays the integral value for a given function. STUDYQUERIES’s triple integral calculator tool makes the calculation faster and displays the integrated value in a fraction of a second.

## How to Use the Triple Integral Calculator?

To use the triple integral calculator, follow these steps:

• Step 1: Input the functions and limits into the appropriate fields
• Step 2: Now, click the “Calculate” button to obtain the integrated value
• Step 3: In the new window, the integrated value is displayed

## What Is Triple Integral?

A triple integral is a combination of 3 successive integrations, used for calculating a volume or for integrating into a fourth dimension, over 3 other independent dimensions.

Examples:

$$\int\int_{0}^{V}\int f(x,y,z)dV$$

$$\int_{1}^{4}\int_{0}^{3x+2}\int_{sin(x+y)}^{cos(x+y)}(2x+e^yz)dz dy dx$$

$$\int_{0}^{2\pi}\int_{0}^{2}\int_{-3}^{3}r dz dr d\theta$$

In order to understand triple integrals, you need to have a solid understanding of double integrals, coordinate geometry in three dimensions, and polar (cylindrical) coordinates. The sums of triple integrals are derived from these topics and cannot be solved without them.

In the same way that a single integral over a curve represents an area, a double integral over a curve represents a volume, a triple integral represents a summation in a hypothetical 4th dimension.

To understand this, imagine a slightly different scenario, where the first 3 dimensions are space, space, and time, and that the fourth dimension is the third space dimension: suppose you have an integral of z where z is a function of x, y, and t, and x and y are also functions of t. The volume as a function of t can be obtained by adding z to x and y. This gives us the numerical volume at an instance of time if we plug in a value for t.

Nevertheless, since volume is a function of time, it changes over time. We are, therefore, adding up all the different volumes as a function of t over a period of time if we integrate this result. For example, imagine that a balloon is being inflated. A double integral can be used to calculate the volume of any particular object at any moment in time. When summing up all the volumes over the entire inflation process, we must use a triple integral as described below.

In the same way that a double integral can be used to represent an area, triple integrals can be used to represent a volume. In the triple integral $$\int_{c}^{d}\int_{a}^{b}\int_{0}^{z}f(x,y,z)dz dy dx$$

If f(x,y,z) = 1 then this triple integral become as $$\int_{c}^{d}\int_{a}^{b}Z(x,y) dy dx$$

The volume under the surface is represented by z(x,y).

### How to Solve Triple Integral Problem?

You can integrate with respect to the 3 variables in any order if you choose the limits properly when evaluating a triple integral of f(x,y,z) over a volume V.

If V represents a rectangular box, then x, y, and z will all be constants and the order of integration will not matter.

$$\int_{e}^{f}\int_{c}^{d}\int_{a}^{b}f(x,y,z)dz dy dx = \int_{c}^{d}\int_{a}^{b}\int_{e}^{f}f(x,y,z)dx dz dy = …$$

There are six different possible orders of integration.

Now consider the case where you want to find the triple integral $$\int\int_{0}^{V}\int f(x,y,z)dV$$ where V is the volume under the plane $$z + 2x + 3y = 6$$, in the first quadrant. In addition to the plane whose equation is given above, V in the first quadrant is bounded by the 3 planes x = 0, y = 0, z = 0.

It is evident that the shape of V is a tetrahedron with the vertices (3,0,0), (0,2,0), (0,0,6), and (0,0,0). The integral can be set up by considering the z-direction first. As we move in this direction, we enter at z = 0 and leave at $$z = 6 − 2x − 3y.$$

After we do this, we can move onto the x or y directions. If y, we enter through ? = 0, and leave through y = \frac{-2}{3}x + 2. Thus, x must be given the limits 0 to 3, and our triple integral is

$$\int_{0}^{3}\int_{0}^{\frac{-2x}{3} + 2}\int_{0}^{6-2x-3y}f(x,y,z)dz dy dx$$

### Key Points

• Recognize when a function of three variables can be embedded over a rectangular box.
• You can evaluate a triple integral by expressing it as an iterated integral.
• Understand when a function of three variables is embeddable over a closed and bounded region.
• A triple integral can be simplified by changing the order of integration.
• Calculate the average value of the function of three variables.

### Triple Integral Examples

Example 1: A cube has sides of length 4. The origin should be at one corner, and the adjacent corners should be on the positive x, y, and z axes.

Find the mass of the cube if its density is proportional to its distance from the xy plane.

Solution: The density of the cube is $$f(x,y,z) = Kz$$ for some constant K.

If W is the cube, the mass is the triple integral

$$\int\int\int_{W}^{ }(Kz)dV = \int_{0}^{4}\int_{0}^{4}\int_{0}^{4}(Kz)dx dy dz$$

$$= \int_{0}^{4}\int_{0}^{4}(Kz) [x]_{0}^{4}dy dz$$

$$= \int_{0}^{4}\int_{0}^{4}(4Kz) dy dz$$

$$= \int_{0}^{4}(4Kz)[y]_{0}^{4} dz$$

$$= \int_{0}^{4}(16Kz) dz$$

$$= (8K) [z^2]_{0}^{4}$$

$$= 128K$$

The mass of the cube is 128 grams if K=1 gram per cubic cm per cm and distance is in cm.

Example 2: Evaluate the integral $$\int_{0}^{1}\int_{0}^{x}\int_{0}^{1+x+y}f(x,y,z)dz dy dx$$ where f(x,y,z)=1.

Solution: $$\int_{0}^{1}\int_{0}^{x}\int_{0}^{1+x+y}dz dy dx$$

$$= \int_{0}^{1}\int_{0}^{x}[z]_{0}^{1+x+y} dy dx$$

$$= \int_{0}^{1}\int_{0}^{x}(1+x+y) dy dx$$

$$= \int_{0}^{1}[y+yx+\frac{y^2}{2}]_{0}^{x} dx$$

$$= \int_{0}^{1}[x+x^2+\frac{x^2}{2}] dx$$

$$= \int_{0}^{1}[x+\frac{3x^2}{2}] dx$$

$$= [\frac{x^2}{2}+\frac{x^3}{2}]_{0}^{1}$$

$$= [\frac{1}{2}+\frac{1}{2}]$$

$$= 1.$$

## Triple Integral Calculator Cylindrical:

The Triple Integral Calculator Cylindrical is a tool that helps in evaluating triple integrals using cylindrical coordinates. Cylindrical coordinates are a system that represents points in three-dimensional space using a radial distance, an azimuthal angle, and a height. The calculator takes the input of the integrand and the limits of integration and provides the result of the triple integral.

For example, suppose we want to evaluate the triple integral ∫∫∫ f(x, y, z) dV over a region R defined in cylindrical coordinates. We can use the Triple Integral Calculator Cylindrical to compute this integral by entering the appropriate integrand and the limits of integration in terms of cylindrical coordinates.

## Triple Integral Calculator With Steps:

The Triple Integral Calculator With Steps is a tool that not only provides the result of a triple integral but also shows the step-by-step process of evaluating the integral. This calculator is beneficial for students and anyone learning about triple integration as it helps in understanding the methodology behind the calculations.

For instance, if we have a triple integral ∫∫∫ f(x, y, z) dV, the Triple Integral Calculator With Steps will display each step involved in evaluating the integral, such as setting up the limits of integration, converting to the appropriate coordinate system if necessary, performing the integrations, and simplifying the result.

## Triple Integral Cylindrical Coordinates Calculator:

The Triple Integral Cylindrical Coordinates Calculator is a specific tool designed to compute triple integrals using cylindrical coordinates. Cylindrical coordinates consist of a radial distance, an azimuthal angle, and a height, which are used to represent points in three-dimensional space. This calculator allows users to input the integrand and the limits of integration in terms of cylindrical coordinates, and it provides the result of the triple integral.

For example, if we have a triple integral ∫∫∫ f(x, y, z) dV over a region R defined in cylindrical coordinates, the Triple Integral Cylindrical Coordinates Calculator enables us to calculate the integral by entering the appropriate function and the limits of integration in terms of cylindrical coordinates.

## Polar Triple Integral Calculator:

The Polar Triple Integral Calculator is a tool used to evaluate triple integrals using polar coordinates. Polar coordinates involve a radial distance and an angle, and they are commonly used to describe points in a plane. This calculator allows users to input the integrand and the limits of integration in terms of polar coordinates, and it provides the result of the triple integral.

For instance, if we have a triple integral ∫∫∫ f(x, y, z) dV over a region R defined in polar coordinates, the Polar Triple Integral Calculator allows us to compute the integral by entering the appropriate integrand and the limits of integration in terms of polar coordinates.

## Cylindrical Coordinates Triple Integral Calculator:

The Cylindrical Coordinates Triple Integral Calculator is a tool that assists in evaluating triple integrals using cylindrical coordinates. Cylindrical coordinates consist of a radial distance, an azimuthal angle, and a height, and they are used to represent points in three-dimensional space. This calculator takes the input of the integrand and the limits of integration in terms of cylindrical coordinates and provides the result of the triple integral.

For example, if we want to evaluate the triple integral ∫∫∫ f(x, y, z) dV over a region R defined in cylindrical coordinates, the Cylindrical Coordinates Triple Integral Calculator allows us to calculate the integral by entering the appropriate integrand and the limits of integration in terms of cylindrical coordinates.

## Triple Integral Calculator Spherical:

The Triple Integral Calculator Spherical is a tool used for evaluating triple integrals using spherical coordinates. Spherical coordinates consist of a radial distance, an

azimuthal angle, and a polar angle and they are used to represent points in three-dimensional space. This calculator enables users to input the integrand and the limits of integration in terms of spherical coordinates, and it provides the result of the triple integral.

For instance, if we have a triple integral ∫∫∫ f(x, y, z) dV over a region R defined in spherical coordinates, the Triple Integral Calculator Spherical allows us to compute the integral by entering the appropriate function and the limits of integration in terms of spherical coordinates.

## Cylindrical Integral Calculator:

The Cylindrical Integral Calculator is a tool that helps in evaluating integrals using cylindrical coordinates. It can handle both double and triple integrals in cylindrical coordinates. This calculator takes the input of the integrand and the limits of integration in terms of cylindrical coordinates and provides the result of the integral.

For example, if we want to compute the integral ∫∫∫ f(x, y, z) dV over a region R defined in cylindrical coordinates, the Cylindrical Integral Calculator allows us to calculate the integral by entering the appropriate integrand and the limits of integration in terms of cylindrical coordinates.

## Triple Integral Calculator Rectangular Coordinates:

The Triple Integral Calculator Rectangular Coordinates is a tool that helps in evaluating triple integrals using rectangular coordinates. Rectangular coordinates, also known as Cartesian coordinates, are the standard coordinates in three-dimensional space, representing points using x, y, and z values. This calculator enables users to input the integrand and the limits of integration in terms of rectangular coordinates and provides the result of the triple integral.

For example, if we have a triple integral ∫∫∫ f(x, y, z) dV over a region R defined in rectangular coordinates, the Triple Integral Calculator Rectangular Coordinates allows us to compute the integral by entering the appropriate function and the limits of integration in terms of rectangular coordinates.

## FAQs

What is a triple integral used for?

As with the double integral, triple integrals can be used to find the volume and to find mass when the region of interest has a variable density.

Are triple integrals 4D?

As a single integral produces a value of 2D and a double integral a value of 3D, a triple integral produces a value of a higher dimension than 3D, namely 4D.

What does a triple integral tell you?

This triple integral gives the mass of the object and is equal to the sum of the masses of all the infinitesimal boxes in R. There is a double integral in the x-y plane over the region D.

Can Triple Integral be negative?

Generally speaking, f can be negative, so you must consider the signed mass, which means that the mass could be negative somewhere. It’s likely you’ve heard this: $$\int {}{} f(x) dx$$ is the signed area underneath the curve when your integrand is a function.

What does a double integral give you?

Double integrals are a way of integrating over a two-dimensional area. We can use them to compute the volume under a surface, among other things.

What is the difference between double and triple integral?

The double integral is used for integrating over a two-dimensional region, while the triple integral is used for integrating over a three-dimensional region.

What is the relation between triple integrals and volume?

Volume and triple integral are the same. As a rule, integrals are used to measure the area under a curve, whether open or bounded. The volume integral is a particular case of the triple integral. A 3-dimensional object’s volume is calculated by using the triple integral.

Can triple integrals be zero?

Based on the definition of center of mass, your integrals represent the product of mass and its x, y, and z coordinates, respectively. Because spherical shells are uniformly dense, we assume that the center of mass is (0,0,0) and therefore all three integrals are zero.

How To Do Triple Integral With Spherical Coordinates On HP Prime Calculator?

To perform a triple integral with spherical coordinates on the HP Prime calculator, you can follow these steps:

1. Press the “Home” button on the calculator to go to the home screen.
2. Enter the integral symbol (∫) by pressing the “Integral” key.
3. Input the integrand function using the appropriate variables. For example, if the function is f(x, y, z), you can enter it as f(R, θ, φ) where R represents the radial distance, θ is the azimuthal angle, and φ is the polar angle.
4. Specify the limits of integration for each variable (R, θ, and φ) using the appropriate values or expressions.
5. Press the “Enter” key to evaluate the triple integral.
6. The HP Prime calculator will provide the result of the triple integral.

Please note that the specific steps may vary slightly depending on the model and firmware version of the HP Prime calculator you are using. Refer to the user manual or documentation provided with the calculator for more detailed instructions.

How To Do A Triple Integral On HP Prime Graphing Calculator?

To perform a triple integral on the HP Prime graphing calculator, you can utilize the CAS (Computer Algebra System) capabilities of the calculator. Here’s a general procedure:

1. Press the “Home” button on the calculator to go to the home screen.
2. Enter the integral symbol (∫) by pressing the “Integral” key.
3. Input the integrand function using the appropriate variables. For example, if the function is f(x, y, z), you can enter it as f(x, y, z).
4. Specify the limits of integration for each variable (x, y, and z) using the appropriate values or expressions.
5. Press the “Enter” key to evaluate the triple integral.
6. The HP Prime calculator will provide the result of the triple integral.

The HP Prime calculator has a built-in CAS that can handle symbolic computations, allowing you to perform more complex integrals. Make sure you have the latest firmware installed and consult the user manual or documentation for more detailed instructions specific to your calculator model.

Triple Integral Calculator Where Domain Is Z>Xy?

Finding a specific online calculator that evaluates triple integrals with a domain where z > xy might be challenging. However, you can still calculate this triple integral manually using a calculator or computer software that supports symbolic mathematics or numerical integration methods.

To compute the triple integral where the domain is z > xy, you can follow these general steps:

1. Set up the triple integral with the appropriate integrand function, limits of integration, and coordinate system (rectangular, cylindrical, or spherical).
2. Determine the region of integration that satisfies the condition z > xy. This region may need to be split into multiple sub-regions depending on the complexity of the condition.
3. Evaluate the triple integral separately for each sub-region, ensuring that the condition z > xy is satisfied.
4. Sum up the results from each sub-region to obtain the final value of the triple integral.

It’s worth noting that if the condition z > xy introduces complex shapes or non-elementary functions, numerical methods or computer software (such as MATLAB, Mathematica, or Python libraries like SciPy) may be more suitable for approximating the integral. These tools provide advanced numerical integration techniques and programming capabilities to handle more complex calculations.

How To Do A Triple Integral On A Calculator?

Performing a triple integral on a standard scientific calculator without computer algebra capabilities can be challenging due to the complexity of the calculations involved. However, you can approximate the result using numerical integration techniques. Here’s a general approach:

1. Divide the region of integration into smaller sub-regions, if necessary, based on the function and the domain.
2. Approximate the volume element within each sub-region using a simple shape (e.g., rectangular or cylindrical) that represents the region.
3. Evaluate the integrand function at various points within each sub-region and approximate the integral by summing the contributions from each point.
4. Adjust the number of sample points within each sub-region to improve the accuracy of the approximation. More sample points generally result in a more accurate result, but it also increases the computation time.
5. Sum up the contributions from all sub-regions to obtain the final approximation of the triple integral.

Please note that performing triple integrals by hand on a calculator can be time-consuming and prone to errors. For more complex integrals or precise results, computer software with numerical integration capabilities, such as MATLAB, Mathematica, or Python libraries like SciPy, would be more suitable. These tools provide advanced numerical integration techniques and programming capabilities to handle complex calculations efficiently.