Math Problem Statement
Solution
The image shows a question about numerical integration using the trapezium (trapezoidal) rule for the integral:
Three different step sizes are used, and the corresponding approximations for the integral are:
- For , the approximation is 4.9368 (incorrect).
- For , the approximation is 4.9062 (correct).
- For , the approximation is 4.8889 (incorrect).
It appears the trapezium rule was applied, but the approximations may differ slightly due to rounding or calculation errors.
Would you like me to walk through how the trapezium rule is applied to this integral? If so, we can calculate the correct values or explain the errors.
Here are 5 related questions to expand on this:
- How is the trapezium rule generally applied to approximate integrals?
- What are the advantages of using smaller step sizes like ?
- How would Simpson's rule compare to the trapezium rule for this integral?
- Can this integral be solved exactly, or is numerical approximation the best approach?
- How do the results change if a midpoint or other numerical methods are applied?
Tip: For better accuracy in numerical integration, using a smaller step size (i.e., more intervals) typically improves the approximation, though computation becomes more intensive.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Numerical Integration
Trapezium Rule
Exponential Functions
Formulas
Trapezium Rule Formula: ∫_a^b f(x) dx ≈ h/2 * (f(a) + 2 * (sum of middle terms) + f(b))
Exponential Function: f(x) = e^(2x)/x^2
Theorems
Fundamental Theorem of Calculus
Trapezoidal Approximation
Suitable Grade Level
Grades 11-12 or university-level numerical methods course
Related Recommendation
Approximation of ∫ from -1 to 2 of e^(-x) Using the Trapezoidal Rule
Trapezoidal Approximation of ∫ e^x from 1 to 3 using 10 Trapezoids
Trapezoidal Rule Estimation of g(x) from 0 to 1.6 and Absolute Error Approximation
Numerical Integration: Trapezoidal, Midpoint, and Simpson's Rule for ∫ e^x / (9 + x^2)
Approximate Integral of 1/(1+x^2) from 0 to 6 Using Trapezoidal Rule