According to Table 7.2.1, the arc length is obtained by evaluating the integral ${\int }_{{\mathrm{\theta }}_1}^{{\mathrm{\theta }}_2}\sqrt{r^2\+{\left(r\prime \right)}^2}\mathit{\ⅆ}\mathrm{\theta }$ for $r\=1\+\cos \left(\mathrm{\θ}\right)$. The radical in the integrand is easily seen to be

so the complete integration reduces to

Note well that integrating $\cos \left(\mathrm{\θ}\/2\right)$ instead of $\left|\cos \left(\mathrm{\θ}\/2\right)\right|$ would lead to the incorrect answer of zero! An alternative to splitting the integral of the absolute value at $\mathrm{\θ}\=\mathrm{\π}$ would be to invoke the symmetry in the integrand and reduce the interval of integration to $\left[0\,\mathrm{\π}\right]$, with the value of that integral simply being doubled.

A more elegant interactive solution can be obtained if first, the cardioid $r\=1\+\cos \left(\mathrm{\θ}\right)$ is defined as a function $R\left(\mathrm{\θ}\right)$. (The variable $R$ is used to avoid assigning to the "working" variable $r$.)