Triple Integral Calculator
A triple integral calculator computes three-dimensional integrals for finding volumes, mass distributions, and probability densities in 3D space. Essential in physics and engineering, it helps solve problems involving heat transfer, fluid dynamics, and electromagnetic fields by integrating functions over three variables. This digital tool simplifies complex calculations that would otherwise require tedious manual computation.
Calculate Triple Integral
Triple Integral Formula
∭V f(x,y,z) dxdydz = ∫z₁z₂ ∫y₁y₂ ∫x₁x₂ f(x,y,z) dx dy dz
How to Use
1. Enter function using variables x, y, z (e.g., x^2 + y + z)
2. Set integration limits for each variable
3. Click Calculate
4. View result in output box
5. Use Clear to reset inputs
Derivation Process
Triple integrals extend double integrals to three dimensions using Riemann sums. For function f(x,y,z) over region V, we divide space into small cubes ΔV = ΔxΔyΔz. The integral is the limit of Σf(x_i,y_j,z_k)ΔV as cube sizes approach zero, representing the sum of function values over entire 3D space.
| Component | Notation |
|---|---|
| Volume Element | dV = dx dy dz |
| Differential Elements | dx, dy, dz |
| Integration Order | Inside-out: x → y → z |
FAQs
1. What's the importance of triple integrals?
Triple integrals calculate 3D space properties like mass, center of gravity, and moment of inertia. Engineers use them for stress analysis, physicists for quantum probability densities, and meteorologists for atmospheric modeling. They extend 2D integration to volumetric calculations essential in multivariable calculus.
2. Can I solve triple integrals step-by-step?
Yes, the calculator performs iterated integration: first integrate with respect to x (treating y,z as constants), then integrate that result with respect to y, and finally integrate with respect to z. This stepwise approach follows Fubini's Theorem for rectangular coordinates.
3. What functions are supported?
The calculator handles polynomial, trigonometric, and exponential functions. Use * for multiplication, ^ for exponents, and standard math functions (sin, cos, exp). Complex functions may require numerical methods or symbolic computation.