Triple Integral Calculator
Triple Integral Calculator: Accurately calculate lumber board feet from length, width, and thickness for woodworking, construction, and material estimation.
x*y*z, x^2 + y^2 + z^2
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What is Triple Integral Calculator?
A Triple Integral Calculator computes integrals over three-dimensional regions to find quantities like volume, mass (with density), total charge, and average values. It symbolically or numerically evaluates iterated integrals, detects bounds, and handles different orders of integration. In calculus, a triple integral over a region \(D \subset \mathbb{R}^3\) is written as:
\[ \iiint_{D} f(x,y,z)\, dV. \]
For rectangular boxes \(D=[a,b]\times[c,d]\times[e,f]\), Fubini’s Theorem lets us compute via iterated integrals:
\[ \iiint_{D} f(x,y,z)\, dV = \int_{e}^{f}\!\!\int_{c}^{d}\!\!\int_{a}^{b} f(x,y,z)\, dx\,dy\,dz, \]
while for volume alone,
\[ \text{Vol}(D)=\iiint_{D} 1\, dV. \]
About the Triple Integral Calculator
This calculator guides you from region definition to final evaluation. It supports cartesian, cylindrical, and spherical coordinates, shows rule-by-rule steps, and can switch the order of integration when convenient. For mass with density \(\rho(x,y,z)\):
\[ m=\iiint_{D}\rho(x,y,z)\, dV, \qquad \bar{f}=\frac{1}{\text{Vol}(D)}\iiint_{D} f(x,y,z)\, dV. \]
When a change of variables simplifies the region or the integrand, the tool applies the Jacobian:
\[ \iiint_{D} f(x,y,z)\, dV = \iiint_{D'} f\!\big(\mathbf{r}(u,v,w)\big)\,\left|\det\!\left(\frac{\partial(x,y,z)}{\partial(u,v,w)}\right)\right|\, du\,dv\,dw. \]
In cylindrical \((r,\theta,z)\) and spherical \((\rho,\phi,\theta)\) coordinates:
\[ dV = r\, dr\, d\theta\, dz, \qquad dV = \rho^{2}\sin\phi\, d\rho\, d\phi\, d\theta. \]
How to Use this Triple Integral Calculator
- Enter the integrand \(f(x,y,z)\) (e.g., \(f= x^2 + y^2\) or a density \(\rho\)).
- Define the region \(D\): choose Cartesian bounds, or switch to cylindrical/spherical with appropriate limits.
- Select the order of integration (e.g., \(dx\,dy\,dz\), \(dr\,d\theta\,dz\)), or let the tool auto-optimize.
- Click Calculate to see step-by-step evaluation, including substitutions and Jacobians when used.
- Review numerical value, symbolic antiderivatives, and optional plots of the region/bounds.
Example (rectangular box): for \(D=[0,1]\times[0,2]\times[0,3]\),
\[ \text{Vol}(D)=\iiint_{D}1\,dV =\int_{0}^{3}\!\!\int_{0}^{2}\!\!\int_{0}^{1}1\,dx\,dy\,dz =(1)\cdot(2)\cdot(3)=6. \]