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:

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

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

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