Let's go through the problems in the image step-by-step.

Problem 2: Evaluate $\int_0^2 (5 - x) \, dx$ using the Riemann sum.

To evaluate this integral using the Riemann sum, we follow these steps:

1. Define the function and interval: $f(x) = 5 - x \quad \text{on the interval} \quad [0, 2]$

2. Divide the interval into $n$ subintervals: The width of each subinterval, $\Delta x$, is: $\Delta x = \frac{2 - 0}{n} = \frac{2}{n}$

3. Determine $x_k$: The $k$-th subinterval is at: $x_k = 0 + k \cdot \Delta x = \frac{2k}{n}$

4. Set up the Riemann sum: Using the left endpoint, the Riemann sum becomes: $\sum_{k=1}^n f(x_k) \Delta x = \sum_{k=1}^n \left(5 - \frac{2k}{n}\right) \cdot \frac{2}{n}$

5. Simplify the sum: Expand the expression inside the summation: $= \sum_{k=1}^n \left(10 - \frac{4k}{n}\right) \cdot \frac{1}{n} = \frac{2}{n} \sum_{k=1}^n \left(5 - \frac{2k}{n}\right)$

After