Triple Integral Calculator

An online triple integral calculator helps you to determine the triple integrated values of the given function. The cylindrical integral calculator evaluates the triple integral with multiple methods and displays the step-by-step calculations. In this article, you can learn how to evaluating triple integrals and much more.

What is Triple Integral?

In mathematics, the triple integral is same as the single or double integral. Normally, triple integration is used to integrating over the three-dimensional space. Triple integral used to determine the volume like the double integrals. But it also determines the mass, when the volume of any body has variable density. The function can be expressed as:

$$ F (x, y, z) $$

The integral is calculated depending on the notation order and how the certain notation is set up.

For example, the order is dxdydz, then integration on x then y then z. The iterated integrals are figure out from the innermost to outermost.

How to Calculate Triple Integral?

In triple integral, you need to evaluate the integration for three variables with respect to three different variables. You have to study the following module to get the ideas on how to evaluate the triple integral. Follow these steps and determine the functions with triple integral calculator or manually.

• Take a function that have three different variables to figure out the triple integral.
• Firstly, perform the integration with one variable to eliminate the certain variable.
• Now, substitute the obtained values in the expression like upper and lower limit.
• While performing integration on one variable, you have to consider the other variables as constants.
• After eliminating the one variable, you should repeat the process to eliminate the other variables to obtain the answer in constant.

However, an online Integral Calculator allows you to evaluate the integrals of the functions with respect to the variable involved.

Example:

Question: 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, triple integral calculator 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, the tripple integral calculator 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, triple integral calculator 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, triple iterated integral calculator simplify the obtaining values:

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

Then, triple integration calculator 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:

Triple integrals are usually calculated by using cylindrical coordinates than rectangular coordinates. Some equations in rectangular coordinates along with related equations in cylindrical coordinates are listed in Table. The equations become easy as cylindrical integral calculator proceed with solving problems using triple integrals.

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

However, an online Derivative Calculator helps to determine the derivative of the function with respect to a given variable.

How does Triple Integral Calculator Works?

An online triple integrals calculator can find the limit of the sum of the product of a function by follow these steps:

Input:

• First, enter a function with respect to x, y, and z variables.
• If you know the upper and lower limit for variables, then choose the definite and substitute upper and lower limits.
• Apart from this, if you have no idea about the limits of variables, then select the indefinite.
• Hit the Calculate Triple Integral button.

Output:

• The cylindrical integral calculator displays the indefinite and definite integral with step-wise calculations.

FAQ:

Why triple integral can be used?

The triple integral mostly used to determine the mass and volume just like the double integral.

What is the volume integral used for?

In calculus, a volume integral refers to the integral over a three-dimensional domain. It is a special case of multiple integrals. In order to calculate flux densities volume integral most commonly used in physics.

How to find the volume of triple integral?

The ellipsoid volume can be represented as the triple integral that is \( V = ∭_U dxdydz = ∭_U’ abcp^2sin θ dpdφdθ \). By symmetry, you can evaluate the volume of ellipsoid lying in the first octant and multiply the results by 8.

Conclusion:

Use this online triple integral calculator to determine the triple integral of entered functions. Triple integral mostly used to determine the volume and mass for the functions with three different variables to integrate over the given interval.

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.
From the source of Lumen Learning: Double Integrals Over Rectangles, Double Integrals Over Rectangles, Iterated Integrals, Double Integrals Over General Regions.