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 with 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 , we divide the interval into equal subintervals, each of width: Let .
Trapezoidal Rule
The Trapezoidal Rule approximation for is given by: where .
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Calculate for :
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Substitute into the Trapezoidal Rule formula:
Midpoint Rule
The Midpoint Rule approximation for is given by: The midpoints for each subinterval are:
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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
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