Polar Coordinates and Area How would we calculate an area using polar coordinates? Our basic increment of area will be shaped like a slice of pie. The slice of pie shown in Figure 1 has rdθ dθ r Figure 1: A slice of pie with radius r and angle dθ. a piece of a circular arc along its boundary with arc length r dθ. We’ll say that dA equals the area of the slice. How do we express dA in terms of r and θ? The total area of the pie this was sliced from is πr2 . To ﬁnd the area dA we note that the proportion of the total area covered equals the proportion of arc length covered. So: dA πr2 = dA = dA = dθ 2πr r dθ · πr2 2πr 1 2 r dθ 2 This is the basic formula for an increment of area in polar coordinates. We want to use polar coordinates to compute areas of shapes other than circles. In this case r will be a function of θ. The distance between the curve and the origin changes depending on what angle our ray is at. Our center point of reference is the origin; we think of rays emerging from the origin at some angle θ; r(θ) is, roughly, the distance we must travel along that ray to get to the curve. To ﬁnd the area of a shape like this, we break it up into circular sectors with angle Δθ. Since the curve is not a circle the circular sectors won’t perfectly cover the region, so we just approximate the area of a wedge between the curve and the origin by: 1 ΔA ≈ r2 Δθ. 2 If we take the limit as Δθ approaches zero our sum of sector areas will approach r=f(θ) Figure 2: A slice from an oddly shaped pie. the exact area and we get: 1 2 r dθ. 2 This is very similar to letting Δx go to zero in a Riemann sum of rectangle areas. In the limit, we have: � θ2 1 2 A= r dθ. θ1 2 dA = Remember that we’re assuming r is a function of θ.

Polar Coordinates and Area

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