Question 1

What happens if $r$ is negative?

Accepted Answer

If $r$ is negative for a certain angle $	heta$, you plot the point at a distance of $|r|$ from the pole, but in the direction exactly opposite to $	heta$. This means you plot it along the line defined by the angle $	heta + \pi$. For example, the point $(-2, \pi/4)$ is plotted at the same location as $(2, 5\pi/4)$.

Question 2

How do I choose the limits of integration for an area calculation?

Accepted Answer

The limits, $\alpha$ and $\beta$, are the starting and ending angles that trace out the boundary of the desired region. A sketch is vital. For a single closed loop that passes through the pole, the limits are often two consecutive values of $	heta$ for which $r=0$. For symmetrical shapes, you can calculate the area of a smaller part and multiply.

Question 3

Why is the area formula $\frac{1}{2}\int r^2 d	heta$ and not something simpler?

Accepted Answer

This formula comes from summing the areas of tiny sectors, not rectangles. The area of a circular sector with radius $r$ and angle $\Delta	heta$ (in radians) is $\frac{1}{2}r^2 \Delta	heta$. When we make the angle infinitesimally small ($d	heta$), we can integrate this expression to find the total area swept out by the radius as it rotates.

Question 4

Can a point have multiple polar coordinates?

Accepted Answer

Yes, unlike Cartesian coordinates, polar coordinates are not unique. Adding any multiple of $2\pi$ to $	heta$ gives the same point. For example, $(r, 	heta)$ is the same point as $(r, 	heta + 2\pi)$ and $(r, 	heta - 2\pi)$. Also, as mentioned, $(r, 	heta)$ is the same as $(-r, 	heta + \pi)$.

Polar coordinates

What you need to know

Mapping with Distance and Direction