I am trying to find the approximate of the following integral_[0,1]e^(-3x^2) with the n = 4. I have to use the Trapezoidal Rule, Midpoint Rule, and the Simpson's Rule

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:
- \( x_{0.125} \approx f(0.125) = 0.9692\)
-

I am trying to find the approximate of the following integral_[0,1]e^(-3x^2) with the n = 4. I have to use the Trapezoidal Rule, Midpoint Rule, and the Simpson's Rule.

Learn how to approximate the integral of e^(-3x^2) from 0 to 1 using numerical methods: Trapezoidal Rule, Midpoint Rule, and Simpson's Rule, with n=4. This guide includes step-by-step calculations and explanations of each method for accurate approximation.

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