Double integrals in polar coordinates

When converting from Cartesian Coordinates to Polar Coordinates, we say that:

$x = r \cos \theta$
$y = r \sin \theta$

We change all $x$'s and $y$'s into $r$'s and $\theta$'s. We also use these formulas that could be useful for conversions:

$x^2 + y^2 = r^2$
$\sqrt{x^2 + y^2} = r$
$\theta = \tan^{-1} \frac{y}{x}$

Keep in mind that polar coordinates are useful when we come across circles or ellipses.

Suppose we have the following integral with region $D$:

$\int \int_D f(x,y)dA$

Then we can convert it into polar coordinates such that:

$\int \int_D f(x,y)dA = \int^{\theta= \beta }_{\theta = \alpha} \int^{r=g_2(\theta )}_{r=g_1(\theta )} f(r \cos \theta, r \sin \theta ) rdrd\theta$

Why do we have to convert to polar coordinates? Watch the video and find out!