Calculate the curvilinear integral of the first kind $\int_{C}ydl$, where $C: x^2 + y^2 = ax$
Solvement $$x^2 - ax + \frac{a^2}{4} - \frac{a^2}{4} + y^2 = 0$$ $$(x - \frac{a}{2})^2 + y^2 = \frac{a^2}{4}$$ Let's introduce the replacement: $$x - \frac{a}{2} = \frac{a}{2}\cos(t), y = \frac{a}{2}\sin(t), 0 \leq t \leq 2\pi$$ $$x = \frac{a}{2}(\cos(t) + 1), y = \frac{a}{2}\sin(t), 0 \leq t \leq 2\pi$$
$$dl = \sqrt{\frac{a^2}{4}(\cos^2(t) + \sin^2(t))}dt = \sqrt{\frac{a^2}{4}}dt = \frac{a}{2}$$
Let's substitute all this into our original integral. $$\int_{0}^{2\pi}\frac{a}{2}\sin(t)\frac{a}{2}dt = \frac{a^2}{4}\int_{0}^{2\pi}sin(t)dt = 0$$
Please tell me if I solved this curvilinear integral correctly? Unfortunately, I didn’t understand this topic well, can you please suggest resources where I can better understand this?