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Triple Integral

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

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Triple Integral -

Tag(s) : Functions, Symbolic Computation

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# Triple Integral

## Triple Integral Calculator

$$\int\limits_3 \int\limits_2 \int\limits_1 f(var_1,var_2,var_3)$$

### How to calculate a 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.

### 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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