In this tutorial we are going to talk about Polar Coordinates. The idea of polar coordinates is that we are not going to keep track of things in terms of x's and y's anymore. Instead, we are going to keep track of points in terms of the radius r and the angle θ. Every point now will have coordinates in terms of r and θ. We will talk about functions r = f(θ). There are sort of two places that calculus comes in in polar coordinates calculating the area inside a curve and calculating the arc length. So, first, we will introduce two formulas for these calculations and then we will use them in some examples.
drawing lots of polar graphs − they do get easier with time.
that when r is negative, you go to the opposite side of the
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.)
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.
finding areas, you often have to integrate sin²θ
To integrate these, use the half-angle formulas from the section on
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.
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.
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
its 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
Convert the polar equation to Cartesian coordinates : r = 3tanθ
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.