Word Count : 277

---

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: __________ ____ __________ ___ __________ ______ __________.
___ ________ ________ ___ _______ ________ __________ _________ ______ ________.
________ ______ _____ _________ _________ _________ ___ ______ _______ _______ _____.
_____ _____ _______ __________ ________.
________ _____ __________ __________ _________ _________ __________ ______ _________ ______.
________ ___ __________ _____ ________ _______.
________ ________ ________ __________ ________ ________ ___ ______ ______ __________.
_________ _________ _______ _______ ____.
_______ ________ _________ ____ ____.
___ _________ ___ ___ ______ __________ _____ _____ __________.
___ ____ __________ _____ _____ _________ ________ ______.
____ ____ ______ ______ ___ _________ ____ ________ __________ _____ ___ __________.
_______ ________ ___ _____ ________ ______.
__________ ___ ___ _______ ________ _____ ____.
_____ _____ _________ ____ ___ ________ ______ ____.
_______ ___ __________ _____ ________.
_____ ________ ________ _______ ________ _______ __________ _________ ______ __________.
___ ___ _____ _______ ____ ____.
__________ _____ ________ ___ ____ _________ ____.
__________ ____ ___ ___ _____ _____ __________ __________ ____ _______ _________.
________ _________ __________ _____ _____ _________ _____ _____ ___ ______.
______ _____ __________ ________ _____ ________.
__________ ____ ______ ______ _____ ________ ____.
__________ ______ _______ ________ _______.
_______ __________ _______ _________ ________ ___ _____ __________ _______ _______.
____ ________ _______ __________ _______ ________ ____ __________.
_________ _______ ________ ___.
Get Full Answer on WhatsApp