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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

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

### Second Integral 2

 Result Format Automatic Selection Exact Value (when possible) Approximate Numerical Value Scientific Notation

## Integral Calculator over a 2D Domain

 Result Format Automatic Selection Exact Value (when possible) Approximate Numerical Value Scientific Notation

### What is a double integral? (Definition)

A double integral is an integral that applies to a function with 2 variables over a given region of the plane.

From a function with 2 variables $f(x, y)$ and a region $R$ of the 2D plane, the general expression of the double integral is $$\iint_R f(x, y) , dA$$ where $dA$ represents a small infinitesimal area in $R$

### 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 $y \in [0,2]$ $$\int_{0}^{2} \int_{0}^{1} x+y \text{ d}x\text{ d}y = \int_{0}^{2} \frac{1}{2}+y \text{ d}y = 3$$

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/surface calculations via double integration by variable change:

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

### Is it possible to change the order of integration?

Yes, in most cases but not always. Changing the order of the integrations can simplify the calculations, but be careful that the limits are independent of the variables.

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Double Integral on dCode.fr [online website], retrieved on 2024-07-18, https://www.dcode.fr/double-integral

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