1. **State the problem:** We want to evaluate the definite integral $$\int_0^2 5x \, dx$$ using the definition of the integral as a limit of Riemann sums. 2. **Find the width of each subinterval:** The interval is from 0 to 2, so the total length is 2. Dividing into $n$ subintervals, the width of each subinterval is $$\Delta x = \frac{2-0}{n} = \frac{2}{n}.$$ 3. **Find the $i$th endpoint $x_i$:** The $i$th endpoint (right endpoint) is $$x_i = 0 + i \Delta x = \frac{2i}{n}.$$ 4. **Evaluate the function at $x_i$:** The function is $f(x) = 5x$, so $$f(x_i) = 5 \cdot \frac{2i}{n} = \frac{10i}{n}.$$ 5. **Set up the Riemann sum:** $$S_n = \sum_{i=1}^n f(x_i) \Delta x = \sum_{i=1}^n \frac{10i}{n} \cdot \frac{2}{n} = \sum_{i=1}^n \frac{20i}{n^2} = \frac{20}{n^2} \sum_{i=1}^n i.$$ 6. **Use the formula for the sum of the first $n$ integers:** $$\sum_{i=1}^n i = \frac{n(n+1)}{2}.$$ 7. **Substitute and simplify:** $$S_n = \frac{20}{n^2} \cdot \frac{n(n+1)}{2} = \frac{20}{n^2} \cdot \frac{n(n+1)}{2} = \frac{20(n+1)}{2n} = \frac{10(n+1)}{n} = 10 + \frac{10}{n}.$$ 8. **Take the limit as $n \to \infty$ to find the integral:** $$\lim_{n \to \infty} S_n = \lim_{n \to \infty} \left(10 + \frac{10}{n}\right) = 10 + 0 = 10.$$ **Final answer:** $$\int_0^2 5x \, dx = 10.$$