Math Problem Statement

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

Solution

To approximate the integral 01e3x2dx\int_0^1 e^{-3x^2} \, dx with n=4n = 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=4n = 4, we divide the interval [0,1][0, 1] into 44 equal subintervals, each of width: h=104=0.25h = \frac{1 - 0}{4} = 0.25 Let f(x)=e3x2f(x) = e^{-3x^2}.

Trapezoidal Rule

The Trapezoidal Rule approximation for n=4n = 4 is given by: Tn=h2[f(x0)+2i=1n1f(xi)+f(xn)]T_n = \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] where xi=0+ih=0.25ix_i = 0 + i \cdot h = 0.25 \cdot i.

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

    • f(0)=e302=1f(0) = e^{-3 \cdot 0^2} = 1
    • f(0.25)=e3(0.25)20.8226f(0.25) = e^{-3 \cdot (0.25)^2} \approx 0.8226
    • f(0.5)=e3(0.5)20.4724f(0.5) = e^{-3 \cdot (0.5)^2} \approx 0.4724
    • f(0.75)=e3(0.75)20.1738f(0.75) = e^{-3 \cdot (0.75)^2} \approx 0.1738
    • f(1)=e3120.0498f(1) = e^{-3 \cdot 1^2} \approx 0.0498
  2. Substitute into the Trapezoidal Rule formula: T4=0.252[1+2(0.8226+0.4724+0.1738)+0.0498]T_4 = \frac{0.25}{2} \left[ 1 + 2(0.8226 + 0.4724 + 0.1738) + 0.0498 \right] T40.252(1+21.4688+0.0498)=0.251.49370.3734T_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=4n = 4 is given by: Mn=hi=1nf(xi1+xi2)M_n = h \sum_{i=1}^n f \left( \frac{x_{i-1} + x_i}{2} \right) The midpoints for each subinterval [xi1,xi][x_{i-1}, x_i] are:

  • x0.125f(0.125)=0.9692x_{0.125} \approx f(0.125) = 0.9692

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Math Problem Analysis

Mathematical Concepts

Numerical Integration
Definite Integral Approximation

Formulas

Trapezoidal Rule: T_n = (h/2) [f(x_0) + 2 Σ f(x_i) + f(x_n)]
Midpoint Rule: M_n = h Σ f((x_{i-1} + x_i) / 2)
Simpson's Rule: S_n = (h/3) [f(x_0) + 4 Σ f(x_{odd}) + 2 Σ f(x_{even}) + f(x_n)]

Theorems

Simpson's Rule Theorem
Trapezoidal Rule Theorem
Midpoint Rule

Suitable Grade Level

Grades 11-12