To approximate the integral $\int_{2}^{10} \sqrt{x^2 + 3} \, dx$ using the Midpoint Rule with $n = 4$, we follow these steps:

Step 1: Define the Midpoint Rule

The Midpoint Rule is given by:

$M_n = \Delta x \sum_{i=1}^{n} f\left(\frac{x_{i-1} + x_i}{2}\right)$

where:

• $\Delta x = \frac{b - a}{n}$ is the width of each subinterval.
• $f\left(\frac{x_{i-1} + x_i}{2}\right)$ is the function evaluated at the midpoint of each subinterval.

Step 2: Determine the Subinterval Width $\Delta x$

Given:

• $a = 2$ (lower limit)
• $b = 10$ (upper limit)
• $n = 4$

The subinterval width is:

$\Delta x = \frac{10 - 2}{4} = 2$

Step 3: Find the Midpoints

The subintervals are:

1. $[2, 4]$
2. $[4, 6]$
3. $[6, 8]$
4. $[8, 10]$

The midpoints are:

$x_1 = \frac{2 + 4}{2} = 3, \quad x_2 = \frac{4 + 6}{2} = 5, \quad x_3 = \frac{6 + 8}{2} = 7, \quad x_4 = \frac{8 + 10}{2} = 9$

Step 4: Evaluate the Function at Each Midpoint

Substituting each midpoint into the function $f(x) = \sqrt{x^2 + 3}$:

$f(3) = \sqrt{3^2 + 3} = \sqrt{9 + 3} = \sqrt{12}$

$f(5) = \sqrt{5^2 + 3} = \sqrt{25 + 3} = \sqrt{28}$

$f(7) = \sqrt{7^2 + 3} = \sqrt{49 + 3} = \sqrt{52}$

$f(9) = \sqrt{9^2 + 3} = \sqrt{81 + 3} = \sqrt{84}$

Step 5: Approximate the Integral Using the Midpoint Rule

Now, substitute these values into the Midpoint Rule formula:

$M_4 = 2 \left(\sqrt{12} + \sqrt{28} + \sqrt{52} + \sqrt{84}\right)$

Let’s compute this value.The approximation of the integral $\int_{2}^{10} \sqrt{x^2 + 3} \, dx$ using the Midpoint Rule with $n = 4$ is $\boxed{50.2637}$, rounded to four decimal places.

Would you like more details on any part of the calculation? Here are some related questions:

1. How do you derive the Midpoint Rule formula?
2. What are the advantages of using the Midpoint Rule for approximating integrals?
3. How would the result change if a different number of subintervals $n$ were used?
4. How does the Midpoint Rule compare to other numerical integration methods like the Trapezoidal Rule?
5. How can this approximation be improved?

Tip: Always check if increasing the number of subintervals (n) can provide a more accurate approximation of the integral.