# The Polar Coordinate System

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In Polar Coordinate System, the references are a fixed point and a fixed line. The fixed point is called the **pole** and the fixed line is called the **polar axis**. The location of a point is expressed according to its distance from the pole and its angle from the polar axis. The distance is denoted by r and the angle by θ.

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# Length of Arc in Polar Plane | Applications of Integration

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The length of arc on polar plane is given by the formula:

$\displaystyle s = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left( \dfrac{dr}{d\theta} \right)^2} ~ d\theta$

The formula above is derived in two ways. See it here: http://www.mathalino.com/reviewer/integral-calculus/length-arc-polar-pla...

# Length of Arc in XY-Plane | Applications of Integration

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The length of arc in rectangular coordinates is given by the following formulas:

$\displaystyle s = \int_{x_1}^{x_2} \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2} \, dx$ and $\displaystyle s = \int_{y_1}^{y_2} \sqrt{1 + \left(\dfrac{dx}{dy} \right)^2} \, dy$

See the derivations here: http://www.mathalino.com/reviewer/integral-calculus/length-arc-xy-plane-...