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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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 stepbystep calculations.
In mathematics, the triple integral is same as the single or double integral. Normally, triple integration is used to integrating over the threedimensional 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.
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.
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 termbyterm:
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 termbyterm:
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 termbyterm:
$$ ∫4x dz = 4xz $$
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 termbyterm:
$$ ∫3z^2 dz = 3∫z^2 dz $$
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:
$$ X^2 yz (8x + 3yz (2z + 1)) / 24 + constant $$
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 
\( X^2 + y^2 = c^2 \) 
\( Z^2 = c^2 (x^2 + y^2) \) 
\( X^2 + y^2 + z^2 = c^2 \) 
\( Z = c(x^2 + y^2) \) 
The calculator can find the limit of the sum of the product of a function by follow these steps:
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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