Question 3 Calculus

Question

Sketch the circle r = 3 cos θ and cardioids  r = 1 + cos θ on the same axis.

a)         Find the area inside both the circle and cardioid.

b)         Find the arc length of that part of the cardioid outside the circle.

Answer 1

Hi Sarah P.,

As you will see from sketching your two functions, the cardioid lies mostly within the circle. So for part (a), first determine .theta. for which the functions cross. Let's call those intersection points as Cartesian coordinates (a, +/- b). Then integrate the cardioid for the zone x(r)>=a, and the circle for the zone x(r)<=a. You will have to figure how to include that lower boundary for the cardioid portion as a f(theta), but that shouldn't be hard, and how to do polar integration, but you probably already do that? And similarly for the circle, though that might be conceptually easier to do by difference from the total sector of the circle which includes the target area minus the triangular area between the center of the circle and the two function intersection points.

Lastly, for the cardioid arc length: remember that you are integrating along the curve, which has a slope with respect to the <theta> direction (along which <r> is measured out). Therefore, the Pythagorean theorem will be of use, as the differential of arc is a function of r, theta, and d(theta).

-- Cheers, -- Mr. d.