Double Integral

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 $

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

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

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/surface calculations via double integration by variable change:

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

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.