41-Calculus-of-PolarEqB - Calculus of Polar Curves Area...

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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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