Curvy Calculus 2

The parametrisation gives $\int_{0}^{2\pi}(0+0+3(2+2\cos t)2\cos t)dt=12\int_{0}^{2\pi}\cos^2 t dt$ $=\boxed{12}\pi$

$\int_{C} x^2e^{5z}dx+x \cos y dy + 3y dz$

$= \int_{0}^{2 \pi} (x^2e^{5z} \frac{dx}{dt}+x \cos y \frac{dy}{dt} + 3y \frac{dz}{dt}) dt$

$= \int_{0}^{2 \pi} ((0)^2 \cdot e^{5z} \frac{dx}{dt}+0 \cdot \cos y \frac{dy}{dt} + 3y \frac{dz}{dt}) dt$

$= \int_{0}^{2 \pi} 3 (2+2 \cos t) \cdot 2 \cos t dt$

$= 12 \int_{0}^{2 \pi} \cos ^2 t + \cos t dt$

$= 12 \int_{0}^{2 \pi} \frac{1+ \cos 2t}{2} + \cos t dt$

$= 12 | \frac{1}{2} t +\frac{1}{4} \sin 2t + \sin t |_{0}^{2 \pi}$

$= 12 \pi \implies A = \boxed{12}$