A vertical line in rectangular coordinates is given by

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A vertical line in rectangular coordinates is given by x=1x=1. A circle in polar form is given by r=2cos⁡θr=2\cos\theta. Above the xx-axis, these two curves enclose a bounded region. Answer the following. (1) Express the line x=1x=1 in polar coordinates. (2) Determine all values of θ\theta (with 0≤θ≤π20\le \theta \le \frac{\pi}{2}) at which the line and the circle intersect. Show your work. (3) Create an integral (or sum of integrals) whose value equals the exact area of the region that is to the right of the line x=1x=1, inside the circle r=2cos⁡θr=2\cos\theta, and above the xx-axis. (4) Evaluate the integral(s) from part (c) to obtain the exact area of the region.

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