Barrons AP Calculus - David Bock

or f ′(θ) dθ.
See Questions 14, 15, and 44 in the Practice Exercises.
BC ONLY

Region Bounded by Polar Curve

FIGURE N7–6
To 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.

BC ONLY

EXAMPLE 3

Find the area inside both the circle r = 3 sin θ and the cardioid r = 1 + sin θ.

SOLUTION: Choosing an appropriate window, graph the curves on your calculator.

See Figure N7–7, where one half of the required area is shaded. Since 3 sin θ = 1 + sin θ when