To approximate the integral $\int_0^1 e^{-3x^2} \, dx$ with $n = 4$ using the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule, we proceed as follows:

Step 1: Set up and evaluate the function at the needed points

Since we have $n = 4$, we divide the interval $[0, 1]$ into $4$ equal subintervals, each of width: $h = \frac{1 - 0}{4} = 0.25$ Let $f(x) = e^{-3x^2}$.

Trapezoidal Rule

The Trapezoidal Rule approximation for $n = 4$ is given by: $T_n = \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]$ where $x_i = 0 + i \cdot h = 0.25 \cdot i$.

1. Calculate $f(x_i)$ for $i = 0, 1, 2, 3, 4$:

   • $f(0) = e^{-3 \cdot 0^2} = 1$
   • $f(0.25) = e^{-3 \cdot (0.25)^2} \approx 0.8226$
   • $f(0.5) = e^{-3 \cdot (0.5)^2} \approx 0.4724$
   • $f(0.75) = e^{-3 \cdot (0.75)^2} \approx 0.1738$
   • $f(1) = e^{-3 \cdot 1^2} \approx 0.0498$

2. Substitute into the Trapezoidal Rule formula: $T_4 = \frac{0.25}{2} \left[ 1 + 2(0.8226 + 0.4724 + 0.1738) + 0.0498 \right]$ $T_4 \approx \frac{0.25}{2} \cdot (1 + 2 \cdot 1.4688 + 0.0498) = 0.25 \cdot 1.4937 \approx 0.3734$

Midpoint Rule

The Midpoint Rule approximation for $n = 4$ is given by: $M_n = h \sum_{i=1}^n f \left( \frac{x_{i-1} + x_i}{2} \right)$ The midpoints for each subinterval $[x_{i-1}, x_i]$ are:

• $x_{0.125} \approx f(0.125) = 0.9692$