Area Bounded by a Polar Curve

Polar coordinates are a two-dimensional coordinate system, where each point on a plane is determined by a distance from a reference point and an angle from a reference direction.
A polar curve is a shape constructed using the polar coordinate system.
The equation of the polar curve is usually given as r = f(θ), where r is the distance from the origin (or pole) and θ is the angle from the polar axis (usually the x-axis).

To calculate the area bounded by a polar curve, we use the formula Area = ½ ∫ (from α to β) [f(θ)]² dθ. Here, ‘α’ and ‘β’ are the limits of the angle θ, and f(θ) is the polar equation representing the curve.
The integral calculates a ‘swept’ area from angle α to β. The factor of ½ accounts for the fact that the area is being calculated in a polar system rather than Cartesian.

Many polar curves display symmetry which can simplify the calculation of areas.
If a polar curve is symmetric about the initial line (θ = 0), it may be enough to calculate the area for θ = 0 to π/2 and then multiply by 4.
If a polar curve is symmetric about θ = π/2, it may be sufficient to calculate the area for θ = 0 to π/2 and then multiply by 2.
However, these shortcuts can only be used if the symmetry of the curve is known and has been confirmed.

Sometimes, the given polar equation might give negative r-values. When integrating to find the area, it is important to use the absolute value of r, as distance cannot be negative.

Understanding and calculating the area bounded by a polar curve is a vital skill in fields such as physics, engineering, and computer graphics.
In real-world situations this could be used, for example, to calculate the area of a radar ‘sweep’, the capture area of a satellite dish, or the area of a petal in a flower for biological or agricultural applications.