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Let's solve this problem step by step.

### Part a) Estimating the integral using the Trapezoidal Rule

The Trapezoidal Rule is used to approximate the value of a definite integral by dividing the area under the curve into trapezoids. The formula for the Trapezoidal Rule is:

\[
\int_{a}^{b} g(x) \, dx \approx \frac{h}{2} \left[ g(x_0) + 2 \sum_{i=1}^{n-1} g(x_i) + g(x_n) \right]
\]

Where:
- \( h \) is the width of each subinterval (in this case \( h = 0.2 \) as the \(x\)-values are equally spaced by 0.2).
- \( x_0, x_1, \dots, x_n \) are the \(x\)-values from the table.
- \( g(x_0), g(x_1), \dots, g(x_n) \) are the corresponding function values from the table.

From the table:
- \( x = [0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6] \)
- \( g(x) = [12.1, 11.6, 11.3, 11.1, 11.7, 12.2, 12.6, 13.0, 13.2] \)

The number of subintervals \(n = 8\), so we calculate the approximation:

\[
\text{Integral} \approx \frac{0.2}{2} \left[ 12.1 + 2(11.6 + 11.3 + 11.1 + 11.7 + 12.2 + 12.6 + 13.0) + 13.2 \right]
\]

I'll compute this now.

### Part b) Estimating the error using the bound on \( g^{(4)}(x) \)

The error bound for the Trapezoidal Rule is given by:

\[
E_T = -\frac{(b - a)^3}{12n^2} M
\]

Where:
- \( [a, b] = [0, 1.6] \)
- \(n = 8\) is the number of subintervals.
- \( M \) is the maximum value of \( |g^{(4)}(x)| \), which is given as \( M = 5 \) (since \( -5 \leq g^{(4)}(x) \leq 2 \)).

Now, I will calculate both the integral and the error estimate.### Results:
- **Part a)**: The approximate value of the integral using the Trapezoidal Rule is **19.23**.
- **Part b)**: The absolute error involved in this approximation is approximately **0.0267**.

Would you like further details on how these calculations were made or any other clarifications?

Here are 5 related questions for deeper understanding:
1. What other numerical integration methods could be used instead of the trapezoidal rule?
2. How does increasing the number of subintervals affect the accuracy of the Trapezoidal Rule?
3. What is the significance of the fourth derivative \( g^{(4)}(x) \) in the error estimation?
4. How would the approximation change if the intervals were not evenly spaced?
5. What are the conditions under which the Trapezoidal Rule gives the exact value for an integral?

**Tip**: For smoother functions, using more subintervals generally improves the accuracy of numerical integration techniques like the Trapezoidal Rule.

Use the following table of data and the trapezoid rule to estimate the integral from 0 to 1.6 of g(x) dx. If -5 ≤ g^(4)(x) ≤ 2 on the interval [0, 1.6], estimate the absolute error involved in the approximation from Part a.

Learn how to use the trapezoidal rule to estimate the integral of a function using given data points. This problem also covers the estimation of absolute error based on the fourth derivative of the function, suitable for high school calculus students.

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