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Double Integral

Tool to calculate Double Integral. The calculation of two consecutive integral makes it possible to compute areas for functions with two variables to integrate over a given interval.

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Double Integral -

Tag(s) : Functions, Symbolic Computation

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# Double Integral

## Double Integral Calculator

$$\int\limits_2 \int\limits_1 f(1,2) \small{\text{ d}\textit{1}\text{ d}\textit{2}}$$

## Integral Calculator over a 2D Domain

### How to calculate a double integral?

The calculation of double integral is equivalent to a calculation of two consecutive integrals, from the innermost to the outermost.

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

Example: Calculate the integral of $f(x,y)=x+y$ over $x \in [0,1]$ and $y \in [0,2]$ $$\int_{0}^{2} \int_{0}^{1} x+y \text{ d}x\text{ d}y = \int_{0}^{2} \frac{1}{2}y^2+y \text{ d}y = 3$$

Enter the function on dCode with the upper and lower bounds for each variable and the calculator will return the resultat automatically.

It is possible to use variables in the bounds of the integrals:

$$\iint (x+y) \text{ d}x \text{ d}y = \int_0^1 \left( \int_0^{y} (x+y) \text{ d}x \right) \text{ d}y$$

### How to integrate with polar coordinates?

Polar coordinates are useful for performing area calculations via double integration by variable change:

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

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