1.(a) Use the Midpoint Rule with n = 4 subintervals to estimate (b) Compute the integral exactly. 2. For the integral of problem #1: (a) Use the Error Bound to find the bound for the error. (b) Compute the error made when using this estimate.

To address your questions, we'll break down the solution step-by-step for both parts of the problem:

### Problem 1: Approximating the Integral Using the Midpoint Rule

#### 1(a). Using the Midpoint Rule with \( n = 4 \) Subintervals

The **Midpoint Rule** for approximating the integral of a function \( f(x) \) over an interval \([a, b]\) with \( n \) subintervals is given by:

\[
\int_a^b f(x) \, dx \approx M_n = \sum_{i=1}^n f\left( x_i^* \right) \Delta x
\]

where:

- \( \Delta x = \frac{b - a}{n} \) is the width of each subinterval.
- \( x_i^* = \frac{x_{i-1} + x_i}{2} \) is the midpoint of each subinterval.

Please provide the specific integral function \( f(x) \), and the interval \([a, b]\) so that I can calculate the Midpoint Rule approximation for \( n = 4 \).

#### 1(b). Computing the Integral Exactly

To compute the integral exactly, we need the explicit form of the function \( f(x) \) and the interval \([a, b]\). If you provide that information, we can proceed to calculate the exact integral using analytical methods.

### Problem 2: Error Analysis for the Midpoint Rule

#### 2(a). Error Bound for the Midpoint Rule

The error bound for the Midpoint Rule is given by:

\[
E_M = \frac{(b - a)^3}{24n^2} \max_{a \leq x \leq b} |f''(x)|
\]

This requires:

- The second derivative of the function \( f(x) \), \( f''(x) \).
- The interval \([a, b]\).
- \( n = 4 \).

#### 2(b). Computing the Error Made Using the Estimate

The actual error in using the Midpoint Rule can be calculated as the absolute difference between the exact integral value and the approximation:

\[
\text{Error} = \left| \text{Exact Value} - \text{Midpoint Approximation} \right|
\]

If you provide the specific function and interval, we can go through the calculation steps in detail and compute the error.

Would you like to proceed with a specific function and interval, or do you need further clarification on any step?

### Related Questions:
1. What is the formula for the Trapezoidal Rule and how does it differ from the Midpoint Rule?
2. Why is it important to consider the error bound when approximating integrals?
3. How can we verify that the computed error is within the theoretical error bound?
4. What are some practical applications of numerical integration methods?
5. Can the Simpson's Rule provide a more accurate estimate compared to the Midpoint Rule?

**Tip:** When using the Midpoint Rule, always remember to calculate midpoints precisely and evaluate the function at these points to ensure accuracy.

Explore the Midpoint Rule for numerical integration with a step-by-step explanation. Learn how to estimate integrals using the Midpoint Rule and analyze the error bounds involved. Includes practical examples and calculations.

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