# Double integrals in polar coordinates

### 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

• 1.
Evaluating Double Integrals Using Polar Coordinates
Evaluate the double integral $\int \int_{D} \sqrt{9x^{2} + 9y^{2}} \, dA$ where the region $D$ is between the first quadrant of $\, x^{2} + y^{2} = 1 \,$ and $\, x^{2} + y^{2} = 4$.

• 2.
Evaluate the double integral $\int \int_{D} x - y \, dA \,$ where the region $D$ is the portion of $\, x^{2} + y^{2} = 4 \,$ in the second quadrant.

• 3.
Evaluate the following double integral

$\large \int_{-2}^{2}\int_{0}^{\sqrt{4 - x^{2}}} \sqrt{e^{x^{2}+y^{2}}} \, dydx$

• 4.
Use double integrals to determine the area of the region that is inside $\, r = 3 + 3 \, sin\theta\,$ and outside $\, r = 1 - \, sin\theta$.