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The polar coordinate system is a coordinate system that uses an angle from a given direction as the independent variable and the distance from a given point as the dependent variable. The given point is called the pole, and the given direction from which the angle is measured is called the polar axis.

Polar functions can be converted to parametric functions.

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

Calculus with polar coordinates

Derivatives

Derivatives of polar functions can be found by converting them into parametric functions, then taking the derivative. This yields the formula

$y'=\frac{r'(\theta)\sin(\theta)+r(\theta)\cos(\theta)}{r'(\theta)\cos(\theta)-r(\theta)\sin(\theta)}$

Area under $r(\theta)$

Given the polar function $r(\theta)$ , the area under the function as a Riemann sum is

$\lim_{n\to\infty}\sum_{i=1}^n\frac{r(\theta_i)^2\Delta\theta_i}{2}$

As a definite integral, this would be

$\frac{1}{2}\int\limits_{\alpha}^{\beta}r(\theta)^2d\theta$

This can also be derived by taking a double integral.

$\iint_A dA=\int\limits_{\alpha}^{\beta}\int\limits_0^{r(\theta)}r\,dr\,d\theta=\frac{1}{2}\int\limits_{\alpha}^{\beta}r(\theta)^2d\theta$

Arc length

Given the polar function $r(\theta)$ , the arc length from $\beta$ to $\alpha$ is equal to

$\int\limits_{\alpha}^{\beta}\sqrt{r'(\theta)^2+r(\theta)^2}\,d\theta$