Area of Polar Curve

Given a curve function 𝑟(𝜃) the area bounded by the curve is defined as:

• $\text{Area of Polar Curve}={\lim }_{n\to \infty }{\sum }_{i=0}^n\frac{1}{2}r({𝜃}_i{)}^2𝛥𝜃\text{as Riemann sum}$
• $\text{Area of Polar Curve}=\frac{1}{2}{\int }_a^{b}r(𝜃{)}^2𝛥𝜃\text{as definite integral}$

where:

• 𝜃_𝑖 = 𝑎 + 𝑖·(𝛥𝜃)
• 𝛥𝜃 = (𝑏 - 𝑎) / 𝑛

Intuition

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𝜋 Definition

• 𝜋 = circumference / diameter
• 𝜋  = circumference / 2𝑟

Circumference Formula

• circumference = 2𝜋𝑟

Area of Circle Formula

• Area of Circle = base * height
• Area of Circle = circumference/2 * 𝑟
• Area of Circle = 𝜋𝑟²

Area of a Slice of Circle Formula

• Area of a Slice = Area of Circle * (slice in radians / total-radians)
• Area of a Slice = (𝜋𝑟²) * (𝑑𝜃/2𝜋)
• Area of a Slice = (1/2)𝑟² * 𝑑𝜃

Area of Polar Curve

• $\text{Area of Polar Curve}={\lim }_{n\to \infty }{\sum }_{i=0}^n\frac{1}{2}r({𝜃}_i{)}^2𝛥𝜃\text{as Riemann sum}$
• $\text{Area of Polar Curve}=\frac{1}{2}{\int }_a^{b}r(𝜃{)}^2𝛥𝜃\text{as definite integral}$

where:

• 𝜃_𝑖 = 𝑎 + 𝑖·(𝛥𝜃)
• 𝛥𝜃 = (𝑏 - 𝑎) / 𝑛