Using the Trapezoidal rule with step size $h = 0.5$, evaluate the integral $I = \int_0^1 (3x^2 + 2x) dx$.

Answer Key:

2.125

### Solution: Function $f(x) = 3x^2 + 2x$. $x_0 = 0: f(0) = 0$. $x_1 = 0.5: f(0.5) = 3(0.25) + 2(0.5) = 0.75 + 1.0 = 1.75$. $x_2 = 1.0: f(1.0) = 3(1) + 2(1) = 5.0$. Trapezoidal rule: $I \approx \frac{h}{2} [f(0) + 2 f(0.5) + f(1.0)] = \frac{0.5}{2} [0 + 2(1.75) + 5.0] = 0.25 [3.5 + 5.0] = 0.25 \times 8.5 = 2.125$. Correct: 2.125 (range 2.10 to 2.15).

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