Figure N7–6To find the area A bounded by the polar curve r = f (θ) and the rays θ = α and
θ = β (see Figure N7–6), we divide the region into n sectors like the one shown.
If we think of that element of area, ΔA, as a circular sector with radius r and
central angle Δθ, its area is given by .
Summing the areas of all such sectors yields the area of the entire region:
The expression above is a Riemann Sum, equivalent to this definite integral:.We have assumed above that f (θ) 0 on [α, β]. We must be careful in
determining the limits α and β in (2); often it helps to think of the required area
as that “swept out” (or generated) as the radius vector (from the pole) rotates
from θ = α to θ = β. It is also useful to exploit symmetry of the curve wherever
possible.
The relations between rectangular and polar coordinates, some common polar
equations, and graphs of polar curves are given in the Appendix.