Evaluate $\int_C (x+y) \,\mathrm{d}x$.

Question

I want to evaluate the the following line integral: $$\int_C (x+y) \,\mathrm{d}x,$$ where $C$ is the semicircle starting at $(0,1)$ and ending at $(0,-1)$ as shown below:

Is there a function of $y$ that I can replace it with to be be able to solve this?

Answer 1

You need to use parametric equations,$$ x=\sin t, y= \cos t$$ for $ t$ from $0$ to $\pi $

Substitute these values for $x$ and $y$ and evaluate the integral.

Answer 2

Parameterizing $C$, since the direction is clockwise we have: \begin{equation} -C=\begin{cases} x(t)=Cost \\ y(t)=Sint \end{cases}, t\in[-\frac{\pi}{2},\frac{\pi}{2}] \end{equation} now, $dx(t)=-Sin(t)$$dt$ and for line integral properties:

$\int_{-C}=-\int_{C}$

so we have \begin{equation} -\int_{-C}(x+y)dx=-\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(Cost+Sint)(-Sint)dt=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(Cost+Sint)(Sint)dt=\frac{\pi}{2} \end{equation}