(BC only) Polar coordinates, derivatives and areas

Learning objectives

Convert between polar and Cartesian coordinates using x = r cos θ and y = r sin θ, and sketch curves such as circles, roses, cardioids and limaçons given as r = f(θ)

Differentiate in polar form by treating x and y as functions of θ, giving dy/dx = (r′ sin θ + r cos θ)/(r′ cos θ − r sin θ), and use it to find horizontal and vertical tangents

Interpret dr/dθ as the rate at which the distance from the pole changes, and distinguish it from dy/dx, the slope of the tangent line

Calculate the area enclosed by a single polar curve as ½∫r² dθ, selecting limits that trace the region exactly once and watching for curves traced more than once over the full turn

Find the area of the region inside one polar curve and outside another as ½∫(r_outer² − r_inner²) dθ, after finding the values of θ at which the curves intersect

Split a polar area calculation into separate integrals when the outer curve changes, and use technology to evaluate the resulting integrals in calculator-active questions