Math Problem Statement
Find the approximations T10 and M10 for 2
1 17 e1⁄x dx. (Round your answers to six decimal places.)
Solution
You are being asked to find two approximations, (the Trapezoidal Rule with ) and (the Midpoint Rule with ) for the definite integral:
Step 1: Trapezoidal Rule Approximation ()
The Trapezoidal Rule is given by the formula: where , , , and , with .
- Compute .
- The values of for are , or equivalently:
- The trapezoidal approximation becomes: where .
Step 2: Midpoint Rule Approximation ()
The Midpoint Rule is given by the formula:
- Again, with , , and , we have .
- The midpoints are the values , which for , become:
- The midpoint approximation becomes: where .
Step 3: Numerical Calculation
Now, we substitute the function values for into the formulas for both and .
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Math Problem Analysis
Mathematical Concepts
Numerical Integration
Trapezoidal Rule
Midpoint Rule
Definite Integral
Formulas
Trapezoidal Rule: T_n = (b - a) / (2n) [f(a) + 2 * Σ f(x_i) + f(b)]
Midpoint Rule: M_n = (b - a) / n * Σ f((x_(i-1) + x_i) / 2)
Theorems
Numerical approximation for definite integrals
Suitable Grade Level
College Calculus or Advanced High School Calculus
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