Phong Do
Calculus of Polar Curves - Area
Finding the area of a polar curve often demands a good sketch of the curve. Especially when you have area enclosed
by 2 different curves where the intersections need to be found. The graph also reveals symmetry which can be very
helpful. We can start formulating the area of a polar curve using the generic area formula for parametric equation
but there is actually a easier way.
Back in
Cal I
, the area was derived from using a rectangular element.
Look at a rectangular element
shown here and the area is simply
( )
f x dx
. You then proceed to construct the Riemann sum which
will become the definite integral. You should be very familiar with this concept.
The points on a polar curve are not defined by the rectangular (Cartesian) grid but they are defined
by a radial distance from the origin
r
and an angular measurement
θ
. If you give this some thought,
it is clear that an area region is “fanned” out
[see the diagram below]
. Taking one individual “fan”
out and it is a
sector
as shown.
Look at any Geometry book and you see that the area of circular sector is given as:
2
1
2
Area
r
θ
=
(Note:
θ is in radian)
So, what is the conceptual approach? Look at a generic polar curve as shown. The area of the region is formed by a
series of these sectors. Each sector has an equal incremental angular measure of
∆
θ
. By adding together all of the
individual sectors (i.e., creating a
Riemann sum)
, you will have the area defined by this polar curve.
In other words:
θ
∆
≈
∑
2
2
1
r
A
The area integral is thus defined as:
θ
θ
θ
d
r
A
∫
=
2
1
2
2
1
As you can see, the above approach is virtually identical to the definite integral that you encountered in Cal I. Instead
of using rectangular element, we use a circular sector element. In using the above rule, pay attention to the factor
½
.
It is
part of the formula
because it comes with the sector area formula. It does not mean that you are finding half of
the area.

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