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 sectoras shown. Look at any Geometry book and you see that the area of circular sector is given as: 212Arearθ=(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∆≈∑221rAThe area integral is thus defined as: θθθdrA∫=21221As 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 formulabecause it comes with the sector area formula. It does not mean that you are finding half of the area.