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
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.