Approximate ∫0to6 1/1+x^2 𝑑𝑥 using
(i). Trapezoidal rule

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.

Approximate ∫0to6 1/1+x^2 dx using (i) Trapezoidal rule

Learn how to approximate the integral ∫0to6 1/(1+x^2) dx using the Trapezoidal Rule with a step-by-step guide. This method is suitable for advanced high school and undergraduate students, providing an accurate approximation using numerical integration techniques.

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