Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 15 / c

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- Solution c

Solution

Differentiate

x = r cos θ

,

y = r sin θ

:

d x = cos θ d r - r sin θ d θ, d y = sin θ d r + r cos θ d θ .

Squaring and adding cancels the mixed terms and uses

cos^{2} θ + sin^{2} θ = 1

d x^{2} + d y^{2} = d r^{2} + r^{2} d θ^{2} .

For this cardioid, the polar arc length integrand is

sin^{2} θ + (1 + cos θ)^{2} = 2 + 2 cos θ = 2∣ cos (θ /2) ∣.

The cosine inside the absolute value changes sign at

θ = π

. Therefore the perimeter is

ℓ = 2 \int_{0}^{π} cos (θ /2) d θ - 2 \int_{π}^{2 π} cos (θ /2) d θ = 8

in coordinate length units. The isolated cusp does not obstruct this piecewise smooth arc length integral.

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