To approximate the integral of $\sqrt[4]{1 + x^2} \, dx$ from 0 to 2 using Simpson's Rule, Trapezoidal Rule, and Midpoint Rule, we need to calculate the approximate values of the integral with $n = 8$ subintervals.

The steps for each method are outlined below:

Step 1: General Setup

We are given: $f(x) = \sqrt[4]{1 + x^2}$ and we need to approximate the integral: $\int_0^2 \sqrt[4]{1 + x^2} \, dx$ with $n = 8$ subintervals.

The interval is from $a = 0$ to $b = 2$, and we divide it into $n = 8$ subintervals. The width of each subinterval $h$ is: $h = \frac{b - a}{n} = \frac{2 - 0}{8} = 0.25$

Step 2: Trapezoidal Rule

The Trapezoidal Rule formula is: $T_n = \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]$ where $x_i = a + ih$ for $i = 0, 1, 2, \dots, n$.

Step 3: Midpoint Rule

The Midpoint Rule formula is: $M_n = h \sum_{i=1}^{n} f\left( \frac{x_{i-1} + x_i}{2} \right)$ where $x_i$ are the endpoints of the subintervals.

Step 4: Simpson's Rule

Simpson's Rule formula is: $S_n = \frac{h}{3} \left[ f(x_0) + 4 \sum_{i=1,3,5,\dots}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,\dots}^{n-2} f(x_i) + f(x_n) \right]$

I will now calculate the approximations using these methods.Here are the approximations for the integral $\int_0^2 \sqrt[4]{1 + x^2} \, dx$ using $n = 8$ subintervals, with results rounded to 6 decimal places:

1. Trapezoidal Rule: $T_n = 2.413790$
2. Midpoint Rule: $M_n = 2.411453$
3. Simpson's Rule: $S_n = 2.412232$

Would you like more details or have any questions?

Here are five related questions you could explore next:

1. How does increasing $n$ affect the accuracy of these approximations?
2. What is the exact value of the integral and how close are these approximations to it?
3. How do these rules compare for other functions, such as exponential or trigonometric functions?
4. Can you use adaptive methods to improve the accuracy for specific intervals?
5. How do Simpson's Rule and Trapezoidal Rule converge compared to the Midpoint Rule for this integral?

Tip: Simpson's Rule often provides a more accurate approximation than the Trapezoidal and Midpoint rules for smooth functions because it uses quadratic interpolations.