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To help you with the trapezoidal rule question in the image, let's break it down step by step.

### Given Problem:

You are asked to estimate the integral

\[
\int_0^1 8 \cos(x^2) \, dx
\]

using the **Trapezoidal Rule** with \( n = 4 \) subintervals.

### Trapezoidal Rule Formula:

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

where:

- \( \Delta x = \frac{b - a}{n} \) is the width of each subinterval.
- \( x_i = a + i \Delta x \) are the points where the function is evaluated.

### Step 1: Determine the values of \( a \), \( b \), and \( \Delta x \).

From the integral:

- \( a = 0 \)
- \( b = 1 \)
- \( n = 4 \)

Now calculate \( \Delta x \):

\[
\Delta x = \frac{b - a}{n} = \frac{1 - 0}{4} = \frac{1}{4} = 0.25
\]

### Step 2: Determine the \( x_i \)'s

Using the formula \( x_i = a + i \Delta x \), we calculate the values of \( x_0, x_1, x_2, x_3, x_4 \):

- \( x_0 = 0 \)
- \( x_1 = 0 + 1 \cdot 0.25 = 0.25 \)
- \( x_2 = 0 + 2 \cdot 0.25 = 0.5 \)
- \( x_3 = 0 + 3 \cdot 0.25 = 0.75 \)
- \( x_4 = 0 + 4 \cdot 0.25 = 1 \)

### Step 3: Apply the function \( f(x) = 8 \cos(x^2) \).

Now, we evaluate \( f(x) = 8 \cos(x^2) \) at each of the \( x_i \)'s:

- \( f(x_0) = 8 \cos(0^2) = 8 \cos(0) = 8 \times 1 = 8 \)
- \( f(x_1) = 8 \cos(0.25^2) = 8 \cos(0.0625) \)
- \( f(x_2) = 8 \cos(0.5^2) = 8 \cos(0.25) \)
- \( f(x_3) = 8 \cos(0.75^2) = 8 \cos(0.5625) \)
- \( f(x_4) = 8 \cos(1^2) = 8 \cos(1) \)

### Step 4: Calculate the Trapezoidal Approximation.

Substitute these values into the trapezoidal rule formula:

\[
T_4 = \frac{0.25}{2} \left[ 8 + 2f(x_1) + 2f(x_2) + 2f(x_3) + 8 \cos(1) \right]
\]

At this point, you would need to compute the values for \( \cos(0.0625) \), \( \cos(0.25) \), \( \cos(0.5625) \), and \( \cos(1) \), and then sum them up to find the trapezoidal rule approximation.

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Let me know if you need further clarification, or I can continue by calculating these cosines and finishing the solution!

Use the Trapezoidal Rule to estimate the integral ∫_0^1 8cos(x^2) dx with n = 4 subintervals.

Estimate the integral ∫_0^1 8cos(x^2) dx using the Trapezoidal Rule with 4 subintervals. Learn how to apply the trapezoidal rule, break down the subintervals, and calculate values step-by-step. Ideal for Grades 11-12 or college-level calculus students.

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