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We are tasked with estimating the integral \( \int_0^1 12 \cos(x^2) \, dx \) using the Trapezoidal Rule and the Midpoint Rule with \( n = 4 \).

### Step 1: Subdivide the Interval
The interval \([0, 1]\) is divided into \(n = 4\) subintervals, so the width of each subinterval is:
\[
\Delta x = \frac{1 - 0}{4} = 0.25
\]

The points where we will evaluate the function are:

- Trapezoidal Rule: \(x_0 = 0\), \(x_1 = 0.25\), \(x_2 = 0.5\), \(x_3 = 0.75\), \(x_4 = 1\)
- Midpoint Rule: The midpoints of the intervals: \(x_{mid,1} = 0.125\), \(x_{mid,2} = 0.375\), \(x_{mid,3} = 0.625\), \(x_{mid,4} = 0.875\)

### Step 2: Apply Trapezoidal Rule
The Trapezoidal Rule formula is:
\[
T = \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right]
\]
where \(f(x) = 12 \cos(x^2)\).

### Step 3: Apply Midpoint Rule
The Midpoint Rule formula is:
\[
M = \Delta x \left[ f(x_{mid,1}) + f(x_{mid,2}) + f(x_{mid,3}) + f(x_{mid,4}) \right]
\]

### Step 4: Numerical Computations
I will now calculate these values numerically. Let me compute them.The numerical estimates for the integral \( \int_0^1 12 \cos(x^2) \, dx \) using \( n = 4 \) are as follows:

- **Trapezoidal Rule Estimate**: \( T \approx 10.749 \)
- **Midpoint Rule Estimate**: \( M \approx 10.907 \)

### Step 5: Analyze Overestimate or Underestimate
From the graph of the function \( 12 \cos(x^2) \), we observe that the function is concave down on the interval \([0, 1]\) because its second derivative \( \frac{d^2}{dx^2}(12 \cos(x^2)) \) is negative in this region. This means:

- The **Trapezoidal Rule** tends to **underestimate** the integral because it under-approximates areas under concave down curves.
- The **Midpoint Rule** tends to **overestimate** the integral because it over-approximates the area for concave down curves.

### Conclusion:
Since the Trapezoidal Rule provides an underestimate and the Midpoint Rule provides an overestimate, the true value of the integral lies between these two estimates. Therefore, we can conclude:
\[
10.749 < \int_0^1 12 \cos(x^2) \, dx < 10.907
\]

Would you like more details on any part of the calculation?

Here are 5 related questions you might find useful:
1. How do you compute the error bounds for the Trapezoidal Rule and Midpoint Rule?
2. How would increasing \( n \) affect the accuracy of these estimates?
3. Can Simpson’s Rule be applied to this problem for a more accurate estimate?
4. What is the exact value of this integral, and how does it compare with these estimates?
5. How does the concavity of a function affect the accuracy of different numerical integration methods?

**Tip**: Always graph the function before applying numerical integration techniques to better understand the behavior of the curve and the expected accuracy of the methods.

Estimate ∫[0 to 1] 12 cos(x^2) dx using the Trapezoidal Rule and the Midpoint Rule, each with n = 4. From a graph of the integrand, decide whether your answers are underestimates or overestimates. What can you conclude about the true value of the integral?

Estimate the integral ∫[0 to 1] 12 cos(x^2) dx using both the Trapezoidal and Midpoint Rule with n = 4. Understand the concavity effect on numerical integration methods and conclude the true value of the integral. Suitable for advanced calculus students.

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