To solve this integral using the definition of the Riemann sum, we start with the given integral:

$I = \int_0^2 (5x^2 - 2x) \, dx$

Step-by-step process:

1. Divide the interval $[0, 2]$ into $n$ subintervals of equal width: $\Delta x = \frac{b-a}{n} = \frac{2-0}{n} = \frac{2}{n}.$

2. Choose sample points $x_i^*$ within each subinterval:

   • For a right Riemann sum, we use $x_i^* = a + i \Delta x = 0 + i \frac{2}{n} = \frac{2i}{n}$, where $i = 1, 2, \dots, n$.

3. Express the function $f(x)$ evaluated at $x_i^*$: $f\left(x_i^*\right) = 5\left(\frac{2i}{n}\right)^2 - 2\left(\frac{2i}{n}\right).$

   Simplify: $f\left(x_i^*\right) = 5\left(\frac{4i^2}{n^2}\right) - 2\left(\frac{2i}{n}\right) = \frac{20i^2}{n^2} - \frac{4i}{n}.$

4. Set up the Riemann sum: $S_n = \sum_{i=1}^n f\left(x_i^*\right) \Delta x.$

   Substitute $f(x_i^*)$ and $\Delta x$: $S_n = \sum_{i=1}^n \left(\frac{20i^2}{n^2} - \frac{4i}{n}\right) \cdot \frac{2}{n}.$

   Simplify: $S_n = \frac{2}{n} \sum_{i=1}^n \left(\frac{20i^2}{n^2} - \frac{4i}{n}\right) = \frac{2}{n} \left(\frac{20}{n^2} \sum_{i=1}^n i^2 - \frac{4}{n} \sum_{i=1}^n i\right).$

5. Use summation formulas:

   • $\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$,
   • $\sum_{i=1}^n i = \frac{n(n+1)}{2}$.

   Substitute these into the Riemann sum: $S_n = \frac{2}{n} \left(\frac{20}{n^2} \cdot \frac{n(n+1)(2n+1)}{6} - \frac{4}{n} \cdot \frac{n(n+1)}{2}\right).$

   Simplify each term:

   • First term: $\frac{20}{n^2} \cdot \frac{n(n+1)(2n+1)}{6} = \frac{20(n+1)(2n+1)}{6n}.$

   • Second term: $\frac{4}{n} \cdot \frac{n(n+1)}{2} = \frac{4(n+1)}{2} = 2(n+1).$

   Combine: $S_n = \frac{2}{n} \left(\frac{20(n+1)(2n+1)}{6n} - 2(n+1)\right).$

6. Take the limit as $n \to \infty$:

   • Expand and simplify $(n+1)(2n+1)$ in the numerator.
   • Only the highest order terms of $n$ will remain significant.

   After simplification: $\lim_{n \to \infty} S_n = \int_0^2 (5x^2 - 2x) \, dx = \frac{20}{3} - 4 = \frac{8}{3}.$

Final Answer:

$I = \frac{8}{3}.$

Let me know if you would like further clarification or a detailed breakdown of any step!

Related Questions:

1. What is the difference between left, right, and midpoint Riemann sums?
2. How does increasing the number of subintervals improve the approximation of the integral?
3. What is the geometric interpretation of this integral?
4. Can you apply the trapezoidal rule to approximate this integral?
5. What happens if we calculate this integral using symbolic integration directly?

Tip: Always visualize the problem if possible. Graphing the function helps you understand its behavior over the given