Word Count : 327

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We are asked to approximate the integral $$ \int_{0}^{8} (x^2 - x + 3) \, dx $$ using Simpson’s 1/3 rule with 8 sub-intervals. Let’s solve this step by step manually. --- ### Step 1: Simpson’s 1/3 Rule formula For $n$ sub-intervals (even) of width $h$: $$ \int_a^b f(x) dx \approx \frac{h}{3} \Big[ f(x_0) + 4(f(x_1)+f(x_3)+\dots+f(x_{n-1})) + 2(f(x_2)+f(x_4)+\dots+f(x_{n-2})) + f(x_n) \Big] $$ where $x_i = a + i h$ and $h = \frac{b-a}{n}$. --- ### Step 2: Determine $h$ and nodes ___ __________ _____ _______ ____ __________ ________ ____ ___ _____.
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