= Solution
Differentiate $x=r\cos\theta$, $y=r\sin\theta$:
$$
dx=\cos\theta\,dr-r\sin\theta\,d\theta,\qquad
dy=\sin\theta\,dr+r\cos\theta\,d\theta.
$$
Squaring and adding cancels the mixed terms and uses $\cos^2\theta+\sin^2\theta=1$, giving
$$
\boxed{dx^2+dy^2=dr^2+r^2d\theta^2.}
$$
For this <cardioid>, the <polar arc length> integrand is
$$
\sqrt{\sin^2\theta+(1+\cos\theta)^2}
=\sqrt{2+2\cos\theta}=2|\cos(\theta/2)|.
$$
The <cosine> inside the <absolute value> changes sign at $\theta=\pi$. Therefore the perimeter is
$$
\ell=2\int_0^\pi\cos(\theta/2)\,d\theta
-2\int_\pi^{2\pi}\cos(\theta/2)\,d\theta
=\boxed{8}
$$
in coordinate <length> units. The isolated <cusp> does not obstruct this piecewise <smooth> <arc length> integral.
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