To estimate the integral of $g(x)$ from $0$ to $1.6$ using the trapezoid rule, we will use the given data and apply the following formula for the trapezoidal approximation:

$\int_{a}^{b} g(x) dx \approx \frac{h}{2} \left( g(x_0) + 2g(x_1) + 2g(x_2) + \dots + 2g(x_{n-1}) + g(x_n) \right)$

Where:

• $h$ is the width of each interval $h = \frac{b - a}{n}$,
• $x_0, x_1, \dots, x_n$ are the values of $x$ in the data,
• $g(x_0), g(x_1), \dots, g(x_n)$ are the corresponding values of $g(x)$.

Step 1: Applying the Trapezoidal Rule

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 intervals $n = 8$, and the width of each interval is $h = 0.2$.

The trapezoidal rule formula becomes:

$\int_{0}^{1.6} g(x) dx \approx \frac{0.2}{2} \left( 12.1 + 2(11.6 + 11.3 + 11.1 + 11.7 + 12.2 + 12.6 + 13.0) + 13.2 \right)$

I'll calculate this now.The trapezoidal approximation for the integral of $g(x)$ from 0 to 1.6 is approximately $19.23$.

Step 2: Estimating the Absolute Error

The error estimate for the trapezoidal rule is given by:

$|E_T| \leq \frac{(b - a)^3}{12n^2} \max |g''(x)|$

However, in this case, you're provided with information about $g^{(4)}(x)$ (the fourth derivative), not $g''(x)$. The formula for error involving the fourth derivative is used for Simpson's rule, but since the fourth derivative bounds are given, we can deduce the magnitude of the error should be small, as follows:

Given that $-5 \leq g^{(4)}(x) \leq 2$, we can proceed to estimate the absolute error.

The error bound involving the fourth derivative for the Simpson’s rule is:

$|E_T| \leq \frac{(b - a)^5}{2880} \max |g^{(4)}(x)|$

Now, I will calculate this error bound using the provided information about $g^{(4)}(x)$ on the interval [0, 1.6]. The maximum value of $|g^{(4)}(x)|$ is 5.

Let’s calculate the error.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to try a different approach or explore related concepts? Here are some additional questions to expand your understanding:

1. How does the trapezoidal rule compare to Simpson's rule for approximating integrals?
2. What is the significance of the fourth derivative $g^{(4)}(x)$ in error estimation?
3. How would you approximate the integral if fewer data points were available?
4. Could a different step size $h$ affect the accuracy of the trapezoidal approximation?
5. How can you use numerical methods to solve integrals when no formula for the function is provided?

Tip: When approximating integrals numerically, smaller intervals (smaller $h$) often lead to more accurate results, though it increases the computational effort.