Areas of regions bounded by polar curves

 To find the area of a region bounded by a curve in rectangular coordinates, we use the Riemann sum to approximate the area under the curve by using rectangles. In polar coordinates, we are going to use the Riemann sum also to find the area bounded by a curve but instead of using rectangles we will use sectors of a circle. Let's consider the curve defined by the function r = f(θ) where α ≤ θ ⩽ 𝛃. Our goal is to find the area bounded by the curve and the 2 radial lines θ = α and θ =𝛃. 

Let's start by dividing the area into sectors of equal width. We name the width Δθ and it's calculated by using this formula: Δθ = 𝛃 - α/n. Let's find the area of the sectors. They have equal area since their measurement is equal. The area of each sector is used to approximate the area between line segments. We sum the area of the sectors to approximate the total area. Let's find the formula for the area of a sector.

The area of a circle is given by A = 𝝅r². The length of a circle is 360 degrees or 2π. The surface for one radian is A = 𝝅r²/2π = r²/2. The area for a sector of Δθ radians is A =  Δθ r²/2. This represents the area of any sector. Let's call it Aᵢ and substitute r by f(θ). Aᵢ = 1/2 [f(θ)]².

Let's add the areas of all the sectors to approximate the area bounded by the polar curve and the radial lines :

Let's divide the sector in as many subintervals as possible. At some point we approach infinity. The area of the sector is then given by:

Theorem                                                                                                                                                   

                                                                                                                                                 

Suppose f is continuous and non negative on the interval  α ≤ θ ⩽ 𝛃 with 0 ≤ α - 𝛃 ≤ 2𝝅. The area of the region bounded by the graph r = f(θ) between the radial lines θ = α and θ = 𝛃   is: