To write the given integral $\int_{2}^{6} 5x^2 \, dx$ as a limit of summations, we can use the definition of the definite integral as the limit of a Riemann sum.

Step-by-Step Process:

1. Divide the interval $[2, 6]$ into $n$ subintervals: The width of each subinterval will be $\Delta x = \frac{6 - 2}{n} = \frac{4}{n}$.

2. Choose sample points:
   We will use the right endpoints for the sample points. The right endpoint of the $i$-th subinterval is: $x_i = 2 + i \Delta x = 2 + \frac{4i}{n}, \quad i = 1, 2, \dots, n.$

3. Sum the values of the function at these sample points: The Riemann sum for this integral is: $S_n = \sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} 5 \left( 2 + \frac{4i}{n} \right)^2 \cdot \frac{4}{n}.$

4. Simplify the expression: $S_n = \sum_{i=1}^{n} 5 \left( 2 + \frac{4i}{n} \right)^2 \cdot \frac{4}{n}.$ Expanding the square inside the summation: $\left( 2 + \frac{4i}{n} \right)^2 = 4 + 2 \cdot \frac{8i}{n} + \frac{16i^2}{n^2}.$ So, $S_n = \sum_{i=1}^{n} 5 \cdot \left( 4 + \frac{8i}{n} + \frac{16i^2}{n^2} \right) \cdot \frac{4}{n}.$ Now, distribute the constants: $S_n = \sum_{i=1}^{n} \left( \frac{20 \cdot 4}{n} + \frac{40 \cdot i}{n^2} + \frac{80 \cdot i^2}{n^3} \right).$

5. Take the limit: The integral is the limit of the Riemann sum as $n \to \infty$: $\int_2^6 5x^2 \, dx = \lim_{n \to \infty} S_n.$

Final Result:

The limit of the sum is: $\int_2^6 5x^2 \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} \left( \frac{20 \cdot 4}{n} + \frac{40 \cdot i}{n^2} + \frac{80 \cdot i^2}{n^3} \right).$

---

Would you like me to clarify any part of this process, or do you have additional questions on this topic?

Related Questions:

1. How can you compute the value of this integral using the limit?
2. What is the geometric interpretation of a Riemann sum?
3. How would the result change if we used left endpoints instead of right endpoints in the Riemann sum?
4. How do you derive the formula for a Riemann sum for a general function?
5. How can we use this method for integrals with more complex functions?

Tip:

When setting up a Riemann sum, always carefully choose the right or left endpoints to match your integral’s requirements. This can significantly affect the sum's approximation.