Polar coordinates — common mistakes

Always draw a sketch before calculating an area. This helps you identify the correct limits of integration and any symmetries. Forgetting the $\frac{1}{2}$ in the area formula is a very common mistake. Also, be prepared to use double angle identities like $\cos(2\theta) = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$ to integrate terms like $\cos^2\theta$ or $\sin^2\theta$.

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

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 $\theta$ for which $r=0$. For symmetrical shapes, you can calculate the area of a smaller part and multiply.

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

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