- Home
- Multivariable Calculus
- Multiple Integrals

# Double integrals in polar coordinates

- Intro Lesson: a5:11
- Intro Lesson: b11:54

### Double integrals in polar coordinates

#### Lessons

__Notes:__

**Review of 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.

**Double Integrals in Polar Coordinates**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!

- Introduction
**Double Integrals in Polar Coordinates Overview:**a)__Review of Polar Coordinates__- change all $x$'s and $y$'s into $r$'s and $\theta$'s
- $x = r \cos \theta$
- $y = r \sin \theta$
- An Example

b)__Double Integrals in Polar Coordinates__- Convert to $r$'s and $\theta$'s
- Add an extra $r$
- Integrate in terms of $r$ & $\theta$
- An Example