To approximate the integral of $\sqrt{x^2 + 5}$ from $x = 2$ to $x = 10$ using the Midpoint Rule with $n = 4$, we'll follow these steps:

Step-by-Step Solution

1. Determine the interval width ($\Delta x$): $\Delta x = \frac{10 - 2}{4} = 2$

2. Identify the midpoints: For $n = 4$, we divide the interval $[2, 10]$ into 4 equal subintervals:

   • First midpoint: $x_1 = 2 + \frac{\Delta x}{2} = 2 + 1 = 3$
   • Second midpoint: $x_2 = 4 + \frac{\Delta x}{2} = 4 + 1 = 5$
   • Third midpoint: $x_3 = 6 + \frac{\Delta x}{2} = 6 + 1 = 7$
   • Fourth midpoint: $x_4 = 8 + \frac{\Delta x}{2} = 8 + 1 = 9$

3. Evaluate the function at each midpoint: We need to evaluate $f(x) = \sqrt{x^2 + 5}$ at each midpoint.

   • $f(3) = \sqrt{3^2 + 5} = \sqrt{9 + 5} = \sqrt{14} \approx 3.7417$
   • $f(5) = \sqrt{5^2 + 5} = \sqrt{25 + 5} = \sqrt{30} \approx 5.4772$
   • $f(7) = \sqrt{7^2 + 5} = \sqrt{49 + 5} = \sqrt{54} \approx 7.3485$
   • $f(9) = \sqrt{9^2 + 5} = \sqrt{81 + 5} = \sqrt{86} \approx 9.2736$

4. Apply the Midpoint Rule: $M_4 = \Delta x \cdot (f(x_1) + f(x_2) + f(x_3) + f(x_4))$ Substituting the values: $M_4 = 2 \cdot (3.7417 + 5.4772 + 7.3485 + 9.2736)$ Calculating the sum inside the parentheses: $M_4 = 2 \cdot 25.8410 = 51.6820$

Answer

$M_4 \approx 51.6820$

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Related Questions

1. How does increasing $n$ affect the accuracy of the Midpoint Rule?
2. What is the difference between the Midpoint Rule and the Trapezoidal Rule?
3. Can we apply Simpson’s Rule to this integral? Why or why not?
4. How do we determine the error in the Midpoint Rule approximation?
5. What is the exact integral value of $\int_2^{10} \sqrt{x^2 + 5} \, dx$ using analytical methods?

Tip

When using numerical integration, more subintervals (higher $n$) typically yield more accurate results.