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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Triple Integral
Triple Integral Calculator
Answers to Questions (FAQ)
What is a triple integral? (Definition)
The triple integral calculation is equivalent to a calculation of three consecutive integrals from the innermost to the outermost.
How to calculate a triple integral?
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.
How to integrate with polar coordinates?
The cylindrical coordinates are often used to perform volume calculations via a triple integration by changing variables:
$$ \iiint f(x,y,z) \text{ d}x\text{ d}y\text{ d}z = \iiint f(r \cos(\theta), r\sin(\theta), z) r \text{ d}r\text{ d}\theta\text{ d}z $$
How to integrate with spherical coordinates?
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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