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Triple Integral Calculator

Enter your function and select the integral type. The calculator immediately figures out the triple integrated values of variables for the function entered, with the steps shown.

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Limit for x:

Limit for y:

Limit for z:

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Triple integral calculator helps you to determine the triple integrated values of the given function. The cylindrical integral calculator evaluates the triple integrals with multiple methods and displays the step-by-step calculations.

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.

How to Calculate Triple Integral?

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 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.

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, 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:

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

\( X^2 + y^2 = c^2 \)

\( Z^2 = c^2 (x^2 + y^2) \)

\( X^2 + y^2 + z^2 = c^2 \)

\( Z = c(x^2 + y^2) \)

How Does Our Calculator Works?

The 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 Triple integrals displays the indefinite and definite integral with step-wise calculations.

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.

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