Tool to calculate triple Integral. The calculation of three consecutive integrals makes it possible to compute volumes for functions with three variables to integrate over a given interval.

Triple Integral - dCode

Tag(s) : Functions, Symbolic Computation

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The triple integral calculation is equivalent to a calculation of three consecutive integrals from the innermost to the outermost.

Calculate the integrals consecutively, from the inside to the outside.

$$ \iiint f(x,y,z) \text{ d}x\text{ d}y\text{ d}z = \int_{(z)} \left( \int_{(y)} \left( \int_{(x)} f(x,y) \text{ d}x \right) \text{ d}y \right) \text{ d}z $$

__Example:__ Calculate the integral of $ f(x,y,z)=xyz $ over $ x \in [0,1] $, $ y \in [0,2] $ and $ z \in [0,3] $ $$ \int_{0}^{3} \int_{0}^{2} \int_{0}^{1} xyz \text{ d}x\text{ d}y\text{ d}z = \int_{0}^{3} \int_{0}^{2} \frac{y^2,z^2}{8} \text{ d}y\text{ d}z = \int_{0}^{3} \frac{z^2}{2} \text{ d}z = \frac{9}{2} $$

Enter the function to be integrated on dCode with the desired upper and lower bounds for each variable and the calculator automatically returns the result.

The cylindrical coordinates are often used to perform volume calculations via a triple integration by changing variables:

$$ \iiint f(x,y,z) \text{ d}x\text{ d}y\text{ d}z = \iiint f(r \cos(\theta), r\sin(\theta), z) r \text{ d}r\text{ d}\theta\text{ d}z $$

The spherical coordinates are often used to perform volume calculations via a triple integration by changing variables:

$$ \iiint f(x,y,z) \text{ d}x\text{ d}y\text{ d}z = \iiint f(\rho \cos(\theta) \sin(\varphi), \rho \sin(\theta)\sin(\varphi), \rho \cos(\varphi) ) \rho^2 \sin(\varphi) \text{ d}\rho \text{ d}\theta \text{ d}\varphi $$

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Cite as source (bibliography):

*Triple Integral* on dCode.fr [online website], retrieved on 2024-11-05,

integral,triple,function,integration,integrate,calculus,volume,antiderivative,primitive

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