Educator

Polar Coordinates

Main formula:

Area =

Arclength =

Hints and tips:

• Practice drawing lots of polar graphs − they do get easier with time.

• Remember that when r is negative, you go to the opposite side of the graph.

• Remember that each point has many different possible sets of polar coordinates. (This is different from rectangular coordinates, where each point has a unique (x, y) pair.)

• To find the limits, you often have to draw the graph and find out what angle θ makes r = 0, or when the graph comes back to meet itself.

• In finding areas, you often have to integrate sin²θ or cos²θ. To integrate these, use the half-angle formulas from the section on trigonometric integrals.

• To find the area between the two graphs, subtract the two area formulas just as you would with rectangular coordinates. To find the limits, set the two functions equal to each other and solve for the angles θ.

• To remember the arclength formula, it helps to recall that it comes from the distance formula between two points, which in turn comes from the Pythagorean Theorem.

• Don’t make the common algebraic mistake of thinking that reduces to a + b! This is extremely wrong, and your teacher will likely be merciless if you do it

• When it’s feasible, check that your answers make sense. Unlike area integrals in rectangular coordinates, which can be negative if a curve goes below the x-axis, areas and arclengths in polar coordinates should always be positive. You might also be able to check geometrically that the area or length of your curve looks approximately right.