Evaluate triple integrals over rectangular regions to find volume, mass, or average values.
Rectangular Box Limits
Integration Results
60u³
Volume (V)
The volume of the rectangular region E defined by the given bounds.
Width (Δx)5.00
Depth (Δy)4.00
Height (Δz)3.00
Visualization
Region E is a rectangular box spanning the Cartesian intervals.
About this calculator
What is a Triple Integral?
A triple integral ∭_E f(x,y,z) dV is a generalization of double integrals to functions of three variables. When f(x,y,z) = 1, the integral represents the volume of the region E. This calculator focuses on rectangular regions, also known as 'boxes', where the limits for x, y, and z are constant.
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Pro Tips
Fubini's Theorem states that for rectangular boxes, you can integrate in any order (e.g., dxdydz or dzdydx) and get the same result.
If the function f(x,y,z) is the density of an object, the triple integral represents the Total Mass.
Use triple integrals to find the center of mass (Centroid) of a 3D object.
For non-rectangular regions, limits of integration often involve other variables (functions of x or y).
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Fun Facts
"Triple integrals are crucial in electromagnetism for calculating the total electric charge in a 3D space."
"In astrophysics, they are used to model the mass distribution of stars and galaxies."
"Clerk Maxwell used volume integrals to formulate his famous equations that describe light and electricity."
"Volume integrals are the 3D equivalent of the 'Area Under the Curve' in 1D calculus."
Formula: V = ∫_{z₁}^{z₂} ∫_{y₁}^{y₂} ∫_{x₁}^{x₂} 1 dx dy dz = (x₂-x₁)(y₂-y₁)(z₂-z₁)
