Triple Integral

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

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

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:

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