Word Count : 277

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We are asked to evaluate [ \int_{0}^{1} \frac{1}{1 + x^2} , dx ] using the trapezoidal rule with (h = 0.5) and (h = 0.25), and then improve it using Romberg’s method. --- Step 1: Trapezoidal Rule The trapezoidal rule formula is: [ T(h) = \frac{h}{2}\left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right] ] Case 1: (h = 0.5) Nodes: (x_0 = 0, x_1 = 0.5, x_2 = 1) Function values: _____ ______ _____ _________ _________ _____ _________ ____.
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