Area Inside r = 1 + cos(θ)

Calculus 2 · Axiom Academy

Calculate the area enclosed by the cardioid using the polar area formula. Find the total area enclosed by the cardioid . The curve traces out completely once as runs from 0 to , so the area is . The cardioid r = 1 + cos θ. Its cusp sits at the pole; the curve reaches x = 2 at θ = 0. We integrate over one full sweep, θ from 0 to 2π. Nice work — you've computed the area inside a cardioid from start to finish. Here's what carried the solution: Polar area formula: for a polar curve , the enclosed area is . Integration bounds: a cardioid traces out exactly once over , so those are the limits. Expand before integrating: square the radius — — before doing anything else. Power-reducing identity: turns the squared cosine into something you can integrate directly. Periodic terms vanish: and integrate to sines that return to 0 over a full period, leaving only the constant. The area inside the cardioid is . The same recipe — square the radius, reduce the powers, integrate over one full trace — handles limaçons, rose curves, and other polar regions.

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