Solution
= Solution
The <arc length> factor is the <norm> of the tangent:
$$
\left|\frac{d\mathbf r}{dt}\right|=\sqrt{36\sin^26t+36\cos^26t+64}=10.
$$
Hence $ds=10\,dt$ and $\Phi=xy=\cos6t\sin6t=\tfrac12\sin12t$. The scalar <line integral> is therefore
$$
\boxed{\int_\Gamma\Phi\,ds=5\int_0^{\pi/12}\sin12t\,dt=\frac56.}
$$
This uses scalar <arc length>, rather than the oriented displacement used in the vector <line integrals> of part (a).