In this video I go over a general formula for the area of a region bounded by two polar curves. I illustrated this in an example in my last video on Example 2, but this time go over the general case, as well as some cautionary notes. The area between two polar curves can be determined by subtracting the two areas of each separate polar curves. The only problem we may face is that often times there are many different polar coordinate representations of the exact same point. This means that we can’t simply solve for points that two polar curves intersect at by setting them equal and solving for the other variable, because some points intersect at different “times”. I illustrate this concept by graphing the individual points on the cardioid and circle curves from Example as the angle increases. It is clear that while there are three intersection points, only two of the points occur at exactly the same time. Thus to know that there are three intersection points, it is often helpful to graph both curves, and what better way to graph polar curves than by the amazing Desmos calculator!
This is a very important video to illustrate not only the general formula for determining the area between two polar curves, but also some of the important factors to consider when determining the area, so make sure to watch this video!
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Area Bounded by Two Polar Curves
My last video, Example 2, illustrated the procedure for finding the area bounded by two polar curves.
In general, let R be a region, as illustrated in the figure below, that is bounded by curves with polar equations r = f(θ) and r = g(θ), θ = a, θ = b, where f(θ) ≥ 0 and 0 < b - a ≤ 2π.
The area A of R is found by subtracting the area inside r = g(θ) from the area inside r = f(θ):
Caution
The fact that a single point has many representations in polar coordinates sometimes makes it difficult to find all the points of intersection of two polar curves.
For instance, it is obvious from the circle and cardioid of Example 2, shown below, that there are three points of intersection:
https://www.desmos.com/calculator/gkxtghtc6r
However, in Example 2 we solved the equations r = 3sinθ and r = 1 + sinθ and found two such points: (3/2 , π/6) and (3/2 , 5π/6).
The origin is also a point of intersection, but we can't find it by solving the equations of the curves because the origin has no single representation in polar coordinates that satisfies both equations.
Notice that, when represented as (0 , 0) or (0 , π), the origin satisfies r = 3 sinθ and so it lies on the circle.
When represented as (0 , 3π/2), it satisfies r = 1 + sinθ and so it lies on the cardioid.
Think of two points moving along the curves as the parameter value θ increases from 0 to 2π.
On one curve the origin is reached at θ = 0 and θ = π.
On the other curve the origin is reached at θ = 3π/2.
The points don't collide at the origin because they reach the origin at different times, but the curves intersect there nonetheless.
Thus, to find all points of intersection of two polar curves, it is recommended that you draw the graphs of both curves.
It is especially convenient to use a graphing calculator or computer to help with this task.