Triple Integral Calculator

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 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 $ ∫_2^3∫_1^3 ∫_0^1 (x^2 + 3xyz^2 + xyz) $ dxdydz?

Solution:

First, take the inner integral $$ ∫ (x^2 + 3xyz^2 + xyz) dx $$

Integrate term-by-term:

The integral of $ x^n is x^n + 1 / n+1 $ when n ≠ −1:

$$ ∫x^2 dx = x^3 / 3 $$ $$ ∫3 xyz^2 dx = 3yz^2 ∫x dx $$

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

$$ ∫x dx = x^2 / 2 $$

So, the result is:

$ 3 x^2yz^2 / 2 $

$$ ∫xyz dx = yz ∫ x dx $$

The integral of

$ x^n is x^{n+1} / n+1 $ when n ≠ −1:

$$ ∫x dx = x^2 / 2 $$

So, the result is:

$ x^2 yz / 2 $

The result is:

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

Now, simplifies the obtain values:

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

Add the constant of integration:

$$ X^2 (2x + 9yz^2 + 3yz) / 6 + constant $$

The answer is:

$$ X^2 (2x + 9yz^2 + 3yz) / 6 + constant $$

Then we take second integral:

$$ ∫x^2 (x / 3 + yz (3z + 1) / 2) dy $$

$$ ∫ x^2 (x / 3 + yz (3z + 1) / 2) dy = x^2∫(x / 3 + yz(3z + 1) / 2) dy $$

Integrate term-by-term:

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

$$ ∫x / 3 dy = xy / 3 $$

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

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

$$ ∫y dy = y^2 / 2 $$

So, the result is:

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

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

So, the result is:

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

Now, simplify:

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

Add the constant of integration:

$$ X^2y (4x + 3yz (3z + 1)) / 12 + constant $$

The answer is:

$$ X^2y (4x + 3yz (3z + 1)) / 12 + constant $$

At the end, triple integral solver take third integral:

$$ ∫x^2y (4x + 3yz (3z + 1)) / 12 dz $$

$$ ∫x^2y (4x + 3yz (3z + 1)) / 12 dz = x^2y ∫(4x + 3yz(3z + 1))dz / 12 $$

Integrate term-by-term:

$$ ∫4x dz = 4xz $$

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

$$ ∫3yz (3z + 1)dz = 3y ∫ z (3z + 1) dz $$

Rewrite the integrand:

$$ Z (3z + 1) = 3z^2 + z $$

Now, integrates term-by-term:

$$ ∫3z^2 dz = 3∫z^2 dz $$

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

$$ ∫z^2 dz = z^3 / 3 $$

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

$$ ∫ z dz = z^2 / 2 $$

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^2y (4xz + 3y (z^3 + z^2 / 2)) / 12 $

Now, simplify the obtaining values:

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

Then, adds the constant of integration:

$$ X^2yz(8x + 3yz (2z + 1)) / 24 + constant $$

The answer is:

$$ X^2 yz (8x + 3yz (2z + 1)) / 24 + constant $$

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:

$\ x=rcos⁡θ$

$\ y=rsin⁡θ$

$\ z = z$

$\ x^{2}+y^{2} = r^{2}$

The “r” arises from the Jacobian determinant of the coordinate transformation and changes the volume element as:
$\ dV=r dr dθ dz$

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

Comparison of Cylindrical and Rectangular Representations of Quadric Surfaces:

	Circular Cylinder	Circular cone	Sphere	Paraboloid
Cylindrical	R = c	Z = cr	$ R^2 + z^2 = c^2 $	$ Z = cr^2 $
Rectangular	$ X^2 + y^2 = c^2 $	$ Z^2 = c^2 (x^2 + y^2) $	$ X^2 + y^2 + z^2 = c^2 $	$ Z = c(x^2 + y^2) $

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

Reference:

From the source of Wikipedia: Multiple integral, Riemann integrable, Methods of integration, integrating constant functions, Use of symmetry, Normal domains on R2.
From the source of Libre Text: Triple Integrals in Cylindrical Coordinates, Integration in Cylindrical Coordinates, Fubini’s Theorem in Cylindrical Coordinates.