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.
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.
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:
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.
Follow these steps:
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:
$$ X^2 yz (8x + 3yz (2z + 1)) / 24 + constant $$
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:
\(\ x=rcosθ\)
\(\ y=rsinθ\)
\(\ z = z\)
\(\ x^{2}+y^{2} = r^{2}\)
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.
|
Circular Cylinder |
Circular cone |
Sphere |
Paraboloid |
Cylindrical |
R = c |
Z = cr |
\( R^2 + z^2 = c^2 \) |
\( Z = cr^2 \) |
Rectangular |
\( X^2 + y^2 = c^2 \) |
\( Z^2 = c^2 (x^2 + y^2) \) |
\( X^2 + y^2 + z^2 = c^2 \) |
\( Z = c(x^2 + y^2) \) |
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
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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