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Triple Integral Calculator

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.

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Triple Integral Calculator

Our triple integral calculator allows you to evaluate both definite and indefinite triple integrals effortlessly. It can compute mass, volume, center of mass, and other 3D properties. You can set integration limits and choose the order of integration to match your problem.

What Is a Triple Integral?

A triple integral extends single and double integrals into three dimensions. It is used to calculate properties of functions over a 3D region, such as volume, mass, and moments of inertia.

It is generally written as:

∭ f(x, y, z) dV

Where:

f(x, y, z) is a function of three variables.
dV is a small volume element in 3D space.

Specifying the integration order (dx dy dz, dy dz dx, etc.) and the limits is critical for proper evaluation.

Steps to Compute a Triple Integral

Start with a function of three variables to integrate.
Select the order of integration, which can affect the complexity of your calculation.
Integrate the first variable, treating the others as constants.
Substitute the integration limits for that variable.
Repeat the process for the remaining two variables to obtain the final result.

Example

Evaluate the triple integral:

\(\int_0^1 \int_1^3 \int_2^3 (x^2 + 3xyz^2 + xyz) \, dx \, dy \, dz\)

Step 1: Integrate with respect to x

\[ \int (x^2 + 3xyz^2 + xyz) \, dx = \frac{x^3}{3} + \frac{3x^2 yz^2}{2} + \frac{x^2 yz}{2} = \frac{x^2(2x + 9yz^2 + 3yz)}{6} + C \]

Step 2: Integrate with respect to y

\[ \int \frac{x^2(2x + 9yz^2 + 3yz)}{6} \, dy = x^2 \int \left( \frac{2x}{6} + \frac{9yz^2 + 3yz}{6} \right) dy = \frac{x^2y(4x + 3yz(3z + 1))}{12} + C \]

Step 3: Integrate with respect to z

\[ \int \frac{x^2y(4x + 3yz(3z + 1))}{12} dz = \frac{x^2 yz (8x + 3yz(2z + 1))}{24} + C \]

The final result of the triple integral is:

\[ \frac{x^2 yz (8x + 3yz (2z + 1))}{24} + C \]

Using Cylindrical Coordinates

For problems with rotational symmetry around the z-axis, cylindrical coordinates simplify computations. The conversion from rectangular to cylindrical coordinates is:

\(x = r \cos \theta\)
\(y = r \sin \theta\)
\(z = z\)
\(x^2 + y^2 = r^2\)

The volume element becomes:

\(dV = r \, dr \, d\theta \, dz\)

Cylindrical coordinates are particularly useful for circular or rotationally symmetric regions.

Cylindrical vs Rectangular Forms of Quadric Surfaces

	Circular Cylinder	Circular Cone	Sphere	Paraboloid
Cylindrical	r = c	z = cr	r² + z² = c²	z = cr²
Rectangular	x² + y² = c²	z² = c²(x² + y²)	x² + y² + z² = c²	z = c(x² + y²)

How to Use Our Triple Integral Calculator

Enter the function \(f(x,y,z)\) and the integration limits.
Select the desired order of integration from the dropdown menu.
Click “Calculate” to view the indefinite or definite integral along with step-by-step details.

References

Wikipedia: Multiple Integrals
LibreTexts: Triple Integrals in Cylindrical Coordinates

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