To approximate the integral $\int_{0}^{6} \frac{1}{1+x^2} \, dx$ using the Trapezoidal rule, we will follow these steps:

1. Divide the interval $[0, 6]$ into $n$ subintervals of equal width $\Delta x$.
2. Apply the Trapezoidal rule formula to approximate the integral.

The Trapezoidal rule formula is given by: $\int_{a}^{b} f(x) \, dx \approx \frac{\Delta x}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]$

Here, $a = 0$, $b = 6$, and $\Delta x = \frac{b-a}{n} = \frac{6-0}{n} = \frac{6}{n}$.

Let's choose $n = 6$ for simplicity, so $\Delta x = 1$.

The points are $x_0 = 0$, $x_1 = 1$, $x_2 = 2$, $x_3 = 3$, $x_4 = 4$, $x_5 = 5$, and $x_6 = 6$.

The function $f(x) = \frac{1}{1+x^2}$ evaluated at these points:

• $f(x_0) = f(0) = \frac{1}{1+0^2} = 1$
• $f(x_1) = f(1) = \frac{1}{1+1^2} = \frac{1}{2}$
• $f(x_2) = f(2) = \frac{1}{1+2^2} = \frac{1}{5}$
• $f(x_3) = f(3) = \frac{1}{1+3^2} = \frac{1}{10}$
• $f(x_4) = f(4) = \frac{1}{1+4^2} = \frac{1}{17}$
• $f(x_5) = f(5) = \frac{1}{1+5^2} = \frac{1}{26}$
• $f(x_6) = f(6) = \frac{1}{1+6^2} = \frac{1}{37}$

Now, applying the Trapezoidal rule formula: $\int_{0}^{6} \frac{1}{1+x^2} \, dx \approx \frac{1}{2} \left[ 1 + 2 \left( \frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \frac{1}{17} + \frac{1}{26} \right) + \frac{1}{37} \right]$

Calculating inside the brackets: $2 \left( \frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \frac{1}{17} + \frac{1}{26} \right) = 2 \left( 0.5 + 0.2 + 0.1 + 0.0588 + 0.0385 \right) = 2 \left( 0.8973 \right) = 1.7946$

So the approximation becomes: [ \frac{1}{2} \left[ 1 + 1.7946 + \frac{1}{37} \right] = \frac{1}{2} \left[ 1 + 1.7946 + 0.0270 \right] = \frac{1}{2} \left[ 2.8216 \right] = 1.4108 ]

Thus, the approximation of the integral $\int_{0}^{6} \frac{1}{1+x^2} \, dx$ using the Trapezoidal rule with $n=6$ is approximately $1.4108$.

Would you like more details on any step or have any questions?

Here are five related questions to explore further:

1. How does the accuracy of the Trapezoidal rule compare to other numerical integration methods like Simpson's rule?
2. What happens to the approximation as the number of subintervals $n$ increases?
3. Can you derive the exact value of $\int_{0}^{6} \frac{1}{1+x^2} \, dx$ and compare it to the approximation?
4. How would you implement the Trapezoidal rule for this integral in a programming language like Python?
5. What are the error bounds for the Trapezoidal rule, and how can they be estimated for this integral?

Tip: Increasing the number of subintervals $n$ generally improves the accuracy of the Trapezoidal rule approximation.