Revise Polar coordinates for Cambridge 9231 (1.5)

A point $P$ in a plane can be uniquely identified by its Cartesian coordinates $(x, y)$. Alternatively, we can define its position using polar coordinates $(r, \theta)$. Here, $r$ is the direct distance from a fixed point called the pole (which corresponds to the origin in the Cartesian system), and $\theta$ is the angle this line segment makes with a fixed direction called the initial line (corresponding to the positive x-axis). The angle $\theta$ is measured anti-clockwise and is typically given in radians.

Conversion from Polar $(r, \theta)$ to Cartesian $(x, y)$: $x = r \cos \theta$ $y = r \sin \theta$

Conversion from Cartesian $(x, y)$ to Polar $(r, \theta)$: $r = \sqrt{x^2 + y^2}$ $\tan \theta = \frac{y}{x}$

To sketch a polar curve $r=f(\theta)$, we investigate how the distance $r$ from the pole changes as the angle $\theta$ sweeps around. A good strategy is to create a table of values for $r$ at key intervals of $\theta$, identify symmetries, find where the curve passes through the pole (by solving $r=0$), and find the maximum and minimum values of $r$. For example, if the equation only involves $\cos \theta$, the curve will be symmetrical about the initial line (the x-axis).

In Cartesian coordinates, we find area by summing the areas of infinitesimally thin rectangles. In polar coordinates, we sum the areas of infinitesimally thin sectors. The area of a small sector of a circle with radius $r$ and angle $d\theta$ is approximately $\frac{1}{2}r^2 d\theta$. By integrating this expression between two angles, we can find the total area of a region bounded by a polar curve.

The area $A$ of the region bounded by the curve $r=f(\theta)$ and the lines $\theta = \alpha$ and $\theta = \beta$ is given by:

$A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta$