To approximate the integral using the midpoint rule, we use the formula:

$M_n = \Delta x \left[ f\left( \frac{x_0 + x_1}{2} \right) + f\left( \frac{x_1 + x_2}{2} \right) + \cdots + f\left( \frac{x_{n-1} + x_n}{2} \right) \right]$

Where:

• $\Delta x = \frac{b - a}{n}$ is the width of each subinterval,
• $x_0 = a$, $x_n = b$,
• $f(x)$ is the function being integrated, and
• $n$ is the number of subintervals.

Step 1: Define the given function and interval

We are approximating the integral of $f(x) = \sin(\sqrt{x})$ over the interval $[0, 24]$ with $n = 4$.

$\int_0^{24} \sin(\sqrt{x}) \, dx$

Step 2: Compute $\Delta x$

The width of each subinterval is given by:

$\Delta x = \frac{24 - 0}{4} = 6$

So, the subintervals are:

• $x_0 = 0$
• $x_1 = 6$
• $x_2 = 12$
• $x_3 = 18$
• $x_4 = 24$

Step 3: Midpoint Rule Calculation

For each subinterval, we need to evaluate the function at the midpoint of the subinterval. These midpoints are:

• Midpoint between $x_0 = 0$ and $x_1 = 6$ is $\frac{0 + 6}{2} = 3$
• Midpoint between $x_1 = 6$ and $x_2 = 12$ is $\frac{6 + 12}{2} = 9$
• Midpoint between $x_2 = 12$ and $x_3 = 18$ is $\frac{12 + 18}{2} = 15$
• Midpoint between $x_3 = 18$ and $x_4 = 24$ is $\frac{18 + 24}{2} = 21$

Now, we evaluate the function $f(x) = \sin(\sqrt{x})$ at each of these midpoints:

1. $f(3) = \sin(\sqrt{3})$
2. $f(9) = \sin(\sqrt{9}) = \sin(3)$
3. $f(15) = \sin(\sqrt{15})$
4. $f(21) = \sin(\sqrt{21})$

Step 4: Summing the values

Now, calculate the sum:

$M_4 = 6 \left[ \sin(\sqrt{3}) + \sin(3) + \sin(\sqrt{15}) + \sin(\sqrt{21}) \right]$

Let's calculate these values:

• $\sin(\sqrt{3}) \approx \sin(1.732) \approx 0.986$
• $\sin(3) \approx 0.1411$
• $\sin(\sqrt{15}) \approx \sin(3.873) \approx -0.3907$
• $\sin(\sqrt{21}) \approx \sin(4.583) \approx -0.989$

Now, calculate the sum:

$M_4 = 6 \left[ 0.986 + 0.1411 - 0.3907 - 0.989 \right] = 6 \times (-0.2526) \approx -1.5156$

Final Answer:

The approximate value of the integral is:

$\boxed{-1.5156}$

Would you like more details or have any questions? Here are some related questions to expand the concepts:

1. How would the result change if we increased $n$?
2. What are the limitations of the midpoint rule for approximating integrals?
3. How do you compare the midpoint rule with the trapezoidal rule for approximating integrals?
4. How do you derive the error estimate for the midpoint rule?
5. Can you approximate the integral using Simpson's rule for comparison?

Tip: For better accuracy in approximating integrals, try increasing the number of subintervals (i.e., using a larger $n$).