Answer
$2$
Work Step by Step
Step 1. The limit of the Riemann sum leads to a definite integral, and from the expression given in the exercise, we can identify that $f(x)=cos(\frac{x}{2})$ and the interval is $[-\pi,0]$.
Step 2. We can write the integral as
$\int_{-\pi}^0f(x)dx=\int_{-\pi}^0cos(\frac{x}{2})dx$
Step 3. Using substitution, let $u=\frac{x}{2}$, and we have $du=\frac{1}{2}dx$ with $x\to-\pi, u\to -\frac{\pi}{2}$ and $x\to0, u\to0$
Step 4. We have
$\int_{-\pi}^0f(x)dx=2\int_{-\frac{\pi}{2}}^0cos(u)du=2sin(u)|_{-\frac{\pi}{2}}^0=0-2sin(-\frac{\pi}{2})=2$
