Area bounded by a cardioid and a loop

Understanding the Cardioid and Loop

A cardioid is a plane curve created as the trace of a point on the perimeter of a circle that is rolling around a fixed circle of the same radius, without slipping. The equation is r = a(1 + cos θ) or r = a(1 + sin θ).
A loop in polar coordinates can be expressed in various forms, one of which is r = 2a cos 2θ, where ‘a’ is the length related to the size of the loop and ‘θ’ is the angle parameter.

The problem of finding the area bounded by a cardioid and a loop requires calculating individual areas under the two curves and subtracting, based upon the relation between cardioid and loop.
The general formula for finding the area in polar coordinates is 1/2∫r(θ)^2 dθ.
When finding the area bounded by a cardioid and a loop, you might have to evaluate the limits of the integral, to find the angles where the two curves intersect. Those intersections can be found by setting the equations of the two curves equal to each other.

Integration under polar coordinates often requires algebraic manipulation and use of trigonometric identities.
You can use the double angle formula for cos 2θ = 2cos^2 θ - 1 or similar identities to help simplify the integral.

The study of cardioids and loops has implications in various fields, including physics, engineering, signal processing and even art and nature.
The phenomenon of the cardioid shape appears in diverse places such as the paths of planets, fluid dynamics, light reflection, and sound acoustics.