To express the integral $\int_{2}^{4} (5 + x^2) \, dx$ as a limit of Riemann sums using right endpoints, follow these steps:

1. Divide the interval: The interval $[2, 4]$ is divided into $n$ subintervals, each of width: $\Delta x = \frac{4 - 2}{n} = \frac{2}{n}.$

2. Define the right endpoints: The right endpoints of the subintervals are given by: $x_i = 2 + i \Delta x = 2 + i \cdot \frac{2}{n}, \quad \text{for } i = 1, 2, \ldots, n.$

3. Approximate the integral: Using the function $f(x) = 5 + x^2$, the sum of the areas of the rectangles is: $\sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} \left(5 + \left(2 + i \frac{2}{n}\right)^2\right) \frac{2}{n}.$

4. Take the limit: The integral is the limit of this sum as $n \to \infty$: $\int_{2}^{4} (5 + x^2) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} \left(5 + \left(2 + i \frac{2}{n}\right)^2\right) \frac{2}{n}.$

Final Expression:

$\int_{2}^{4} (5 + x^2) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} \left(5 + \left(2 + \frac{2i}{n}\right)^2\right) \frac{2}{n}.$

Would you like more details or have any questions about this? Here are related questions for practice:

1. How do left endpoint Riemann sums differ from right endpoint Riemann sums?
2. How does the formula for $\Delta x$ change if the interval is $[a, b]$?
3. What happens to the Riemann sum if $f(x)$ is constant over the interval?
4. How can the midpoint method be used for approximating integrals?
5. How is a definite integral geometrically interpreted?

Tip: Always sketch the function and the rectangles to visually confirm how the sum approximates the area under the curve.