Triple Integral Calculator

Triple Integral Calculator

Enter the function you want to integrate, specify the integration limits for each variable (x, y, and z), and click “Calculate” to obtain the result.

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Triple Integral Calculator

Use our Triple Integral Calculator to evaluate indefinite and definite triple integrals easily. Determine the mass, volume, center of mass, and more for 3D objects effortlessly. Change the order of integration and set the limit for integration to meet your needs.

What Is a Triple Integral?

In mathematics, the triple integral is the extension of the single or double integral. It is the method of evaluating the triple integration over the three-dimensional space.

Triple integral is used to calculate different properties of an object such as mass, volume, etc.

A triple integral is expressed as:

∭ f(x, y, z) dV

Where:

f(x, y, z) is the function containing three variables
dV represents the small volume within the three-dimensional region

Defining the order of integration (dx dy dz, dy dx dz, etc) and the limit of integration is very necessary for the accurate evaluation of integrals.

How To Calculate Triple Integral?

Follow these steps:

Take a function that has three different variables to figure out the triple integral
Choose the order of integration because the order of integration has a direct impact on the complexity of the calculation
Perform the integration with the first variable and consider the other variables as constants
Now, substitute the limits of integration
After eliminating the one variable, repeat the process to eliminate the other variables to obtain the answer

Example

Solve $ \int_{2}^{3} \int_{1}^{3} \int_{0}^{1}(x^{2} + 3xyz^{2} + xyz) \, dx dy dz $?

Solution:

First, take the inner integral:

$$ \int (x^{2} + 3xyz^{2} + xyz) dx $$

Integrate term-by-term:

The integral of $ x^n $ is $ x^{n+1}/n+1 $ when $ n ≠ -1 $:

$$ \int x^{2} dx = x^{3}/3 $$

$$ \int 3xyz^{2} dx = 3yz^{2} \int x dx $$

The integral of $ x^n $ is $ x^{n+1}/n+1 $ when $ n ≠ -1 $:

$$ \int x dx = x^{2}/2 $$

So, the result is:

$$ 3x^{2}yz^{2}/2 $$

$$ \int xyz dx = yz \int x dx $$

The integral of $ x^{n} $ is $ x^{n+1}/n + 1 $ when $ n ≠ -1 $:

$$ \int x dx = x^{2}/2 $$

So, the result is:

$$ x^{2}yz/2 $$

The result is:

$$ x^{3}/3 + 3x^{2}yz^{2}/2 + x^{2}yz/2 $$

Now, simplifies the obtain values:

$$ x^{2}(2x + 9yz^{2} + 3yz)/6 $$

Add the constant of integration:

$$ x^{2}(2x + 9yz^{2} + 3yz)/6 + \text{constant} $$

The answer is:

$$ x^{2}(2x + 9yz^{2} + 3yz)/6 + \text{constant} $$

Then we take second integral:

$$ \int x^{2}(x/3 + yz(3z + 1)/2) dy $$

$$ \int x^{2}(x/3 + yz(3z + 1)/2) dy = x^{2} \int (x/3 + yz(3z + 1)/2) dy $$

Integrate term-by-term:

The integral of a constant is the constant times the variable of integration:

$$ \int x/3 dy = xy/3 $$

$$ \int yz(3z + 1)/2 dy = z(3z + 1) \int y dy/2 $$

The integral of $ y^{n} $ is $ y^{n+1}/n + 1 $ when $ n ≠ -1 $:

$$ \int y dy = y^{2}/2 $$

So, the result is:

$$ y^{2}z(3z + 1)/4 $$

$$ xy/3 + y^{2}z(3z + 1)/4 $$

So, the result is:

$$ x^{2}(xy/3 + y^{2}z(3z + 1)/4) $$

Now, simplify:

$$ x^{2}y(4x + 3yz(3z + 1))/12 $$

Add the constant of integration:

$$ x^{2}y(4x + 3yz(3z + 1))/12 + \text{constant} $$

The answer is:

$$ x^{2}y(4x + 3yz(3z + 1))/12 + \text{constant} $$

At the end, triple integral solver take third integral:

$$ \int x^{2}y(4x + 3yz(3z + 1))/12 dz $$

$$ \int x^{2}y(4x + 3yz(3z + 1))/12 dz = x^{2}y \int (4x + 3yz(3z + 1)) dz / 12 $$

Integrate term-by-term:

$$ \int 4x dz = 4xz $$

The integral of a constant times a function is the constant times the integral of the function:

$$ \int 3yz(3z + 1) dz = 3y \int z(3z + 1) dz $$

Rewrite the integrand:

$$ z(3z + 1) = 3z^{2} + z $$

Now, integrates term-by-term:

$$ \int 3z^{2} dz = 3 \int z^{2} dz $$

The integral of $ z^{n} $ is $ z^{n+1}/n + 1 $ when $ n ≠ -1 $:

$$ \int z^{2} dz = z^{3}/3 $$

So, the result is: $ z^{3} $ The integral of $ z^{n} $ is $ z^{n+1}/n + 1 $ when $ n ≠ -1 $:

$$ \int z dz = z^{2}/2 $$

The result is: $ z^{3} + z^{2}/2 $

$$ 3y(z^{3} + z^{2}/2) $$

$$ 4xz + 3y(z^{3} + z^{2}/2) $$

So, the result is: $ x^{2}y(4xz + 3y(z^{3} + z^{2}/2))/12 $

Now, simplify the obtaining values:

$$x^{2}yz(8x + 3yz(2z + 1))/24$$

Then, adds the constant of integration:

$$x^{2}yz(8x + 3yz(2z + 1))/24 + \text{constant}$$

The answer is:

$$x^{2}yz(8x + 3yz(2z + 1))/24 + \text{constant}$$

Integration in Cylindrical Coordinates:

The cylindrical coordinates are the system used to show points in the 3-dimensional space, helping to solve problems involving the rotational symmetry around the z-axis. The relationship between the cylindrical and rectangular coordinates is as follows:

$x = r \cos \theta$

$y = r \sin \theta$

$z = z$

$x^2 + y^2 = r^2$

The “r” arises from the Jacobian determinant of the coordinate transformation and changes the volume element as:

$dV = r \, dr \, d\theta \, dz$

Using the cylindrical integral calculator makes problem-solving easy and efficient involving triple integrals.

Cylindrical vs Rectangular Forms of Quadric Surfaces

	Circular Cylinder	Circular Cone	Sphere	Paraboloid
Cylindrical	r = c	z = cr	r² + z² = c²	z = cr²
Rectangular	x² + y² = c²	z² = c²(x² + y²)	x² + y² + z² = c²	z = c(x² + y²)

How To Use Our Triple Integral Calculator?

To use the triple integral calculator online, follow these steps:

Step 1: Input the function f (x,y,z) and the limits of integration in the designated fields of the calculator
Step 2: Choose the order of integration from the drop-down menu
Step 3: Click on the “CALCULATE” button to see the indefinite and definite integral with detailed steps

References

Wikipedia: Multiple Integrals
LibreTexts: Triple Integrals in Cylindrical Coordinates