use the midpoint rule with n=4 subnitervals to estimate the integral of sqrt(x) with bounds 9,1

To estimate the integral \(\int_{1}^{9} \sqrt{x} \, dx\) using the midpoint rule with \(n = 4\) subintervals, follow these steps:

### Step 1: Determine the width of each subinterval
The width of each subinterval, \(\Delta x\), is calculated as:
\[
\Delta x = \frac{b - a}{n} = \frac{9 - 1}{4} = \frac{8}{4} = 2
\]

### Step 2: Determine the midpoints of each subinterval
For \(n = 4\), the subintervals are:

1. \([1, 3]\)
2. \([3, 5]\)
3. \([5, 7]\)
4. \([7, 9]\)

The midpoints \(x_i^*\) of these subintervals are:

- \(x_1^* = \frac{1 + 3}{2} = 2\)
- \(x_2^* = \frac{3 + 5}{2} = 4\)
- \(x_3^* = \frac{5 + 7}{2} = 6\)
- \(x_4^* = \frac{7 + 9}{2} = 8\)

### Step 3: Evaluate the function at each midpoint
Now, evaluate \(\sqrt{x}\) at each midpoint:

- \(\sqrt{2} \approx 1.414\)
- \(\sqrt{4} = 2\)
- \(\sqrt{6} \approx 2.449\)
- \(\sqrt{8} \approx 2.828\)

### Step 4: Apply the midpoint rule formula
The midpoint rule formula is:
\[
\text{Midpoint Approximation} = \Delta x \sum_{i=1}^{n} f(x_i^*)
\]
Substitute the values we calculated:
\[
\text{Midpoint Approximation} = 2 \left(\sqrt{2} + \sqrt{4} + \sqrt{6} + \sqrt{8}\right)
\]

\[
\text{Midpoint Approximation} = 2 \left(1.414 + 2 + 2.449 + 2.828\right)
\]

\[
\text{Midpoint Approximation} = 2 \times 8.691 \approx 17.382
\]

### Conclusion
The midpoint rule approximation for \(\int_{1}^{9} \sqrt{x} \, dx\) using 4 subintervals is approximately **17.382**.

Would you like further details on this process or have any questions?

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**Related Questions:**

1. How does the midpoint rule compare to the trapezoidal rule?
2. What are the advantages of using more subintervals in numerical integration?
3. How would the approximation change if \(n\) were increased to 8?
4. Can you explain the error estimation in the midpoint rule?
5. What other methods can be used to approximate definite integrals?

**Tip:** The accuracy of the midpoint rule improves with an increasing number of subintervals, especially when the function is nonlinear.

Learn how to estimate the integral of sqrt(x) from 1 to 9 using the midpoint rule with n=4 subintervals. Follow step-by-step calculations and understand the application of the midpoint rule formula. Approximate the integral and improve your understanding of numerical integration techniques.

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