The given problem involves approximating integrals using two numerical methods: the Trapezoidal Rule and Simpson’s Rule. Each method provides an approximation for definite integrals when calculating them analytically is difficult or impossible. You are also asked to find the error bounds for both methods.

Let's break this down for each part:

General Formulas for Approximation Methods

1. Trapezoidal Rule: $T_n = \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]$ Where $h = \frac{b-a}{n}$, and $x_i = a + i \cdot h$, with $a$ and $b$ being the bounds of the integral, and $n$ the number of subintervals.

2. Simpson's Rule: $S_n = \frac{h}{3} \left[ f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n) \right]$ This method requires $n$ to be even, and similarly, $h = \frac{b-a}{n}$.

Error Bounds

For each method, the error bound depends on the second or fourth derivative of the function being approximated.

1. Trapezoidal Rule Error: $E_T \leq \frac{(b-a)^3}{12n^2} \max_{x \in [a,b]} |f''(x)|$

2. Simpson's Rule Error: $E_S \leq \frac{(b-a)^5}{180n^4} \max_{x \in [a,b]} |f^{(4)}(x)|$

Now, let’s consider each part from your image:

(a) Approximate $\int_0^{\frac{\pi}{4}} \cos(2x)\,dx$ with $n = 6$

• The function is $f(x) = \cos(2x)$
• Bounds: $a = 0$, $b = \frac{\pi}{4}$
• $n = 6$, so divide the interval into 6 subintervals.

(b) Approximate $\int_1^2 \sqrt{1 - \frac{1}{x}} \,dx$ with $n = 6$

• The function is $f(x) = \sqrt{1 - \frac{1}{x}}$
• Bounds: $a = 1$, $b = 2$
• $n = 6$

(c) Approximate $\int_{-1}^1 e^{-x^2}\,dx$ with $n = 4$

• The function is $f(x) = e^{-x^2}$
• Bounds: $a = -1$, $b = 1$
• $n = 4$

(d) Approximate $\int_0^1 (1 + x^3)^{\frac{1}{3}}\,dx$ with $n = 4$

• The function is $f(x) = (1 + x^3)^{\frac{1}{3}}$
• Bounds: $a = 0$, $b = 1$
• $n = 4$

Would you like me to walk you through the solution for each one individually (Trapezoidal Rule, Simpson's Rule, and error estimates), or would you prefer I proceed with a specific part first?

Here are 5 related questions that expand on this:

1. How is the Trapezoidal Rule derived?
2. What is the difference in accuracy between Trapezoidal and Simpson’s Rule?
3. How do you find the second and fourth derivatives needed for the error bounds?
4. Why is Simpson's Rule generally more accurate than the Trapezoidal Rule for the same number of subintervals?
5. What are the limitations of numerical integration methods like Trapezoidal and Simpson's?

Tip: The more subintervals you use in numerical methods like the Trapezoidal or Simpson's Rule, the more accurate your result becomes, but computational cost also increases.