Find the area of the region that lies inside both a given cardioid and a given circle in polar coordinates.

Vault Of Euclid

Area of the Common Interior of a Cardioid and a Circle

Parametric equations and vector functions core

Find the area of the common interior of \( r = 2 + 2 \cos \theta \) and \( r = 2 \sin \theta \).

The solution and a full step-by-step explanation are here. Sign in to read them.

What you need first

Graphing polar curves
Finding intersections of polar curves
Setting up integrals for area in polar coordinates
Evaluating trigonometric integrals using half-angle identities

What this problem tests

Tests the ability to find intersection points of polar curves, determine which curve bounds the region over specific intervals, and evaluate trigonometric integrals.

This is a standard problem in Calculus II or III when studying polar coordinates and integration.

Area of the Common Interior of a Cardioid and a Circle

What you need first

What this problem tests

More on Parametric equations and vector functions