Math Problem Statement
For questions 3, 4, and 5, you will use b = 2. Enter the value of M6, correct to 3 decimals, into the answer box below.
Solution
To modify the given code to approximate the integral using the midpoint Riemann sum , we need to make a few adjustments in the code. Specifically, we will:
- Set (as specified).
- Update for (to divide the interval into 6 subintervals).
- Change the function definition to .
Here’s the modified code with these updates:
pythonimport numpy as np # Define parameters a = 0 b = 2 n = 6 # Set to 6 for M6 delta_x = (b - a) / n # Compute the base length of each rectangle # Define independent variable x = np.linspace(a, b, n + 1) # x-values at the left and right endpoints of each subinterval x_mid = x[:-1] + delta_x / 2 # Midpoints of each interval # Define function for the integral y_mid = np.exp(-x_mid**3) # Compute the midpoint Riemann sum MS = sum(y_mid) * delta_x # Riemann sum approximation # Output statement print("The midpoint sum M6 is:", round(MS, 3))
Explanation of Changes:
- Setting : This limits the integration to the interval .
- Setting : This gives us 6 subintervals, as required for .
- Defining : The integrand function is updated to match the problem.
This code will calculate , the midpoint Riemann sum approximation for the given integral, and print the result rounded to three decimal places.
Would you like further clarification on any part of this code? Here are some related questions to deepen your understanding:
- How does changing affect the accuracy of a Riemann sum approximation?
- What is the purpose of using midpoint values in this calculation?
- How does the Riemann sum method relate to the actual integral value?
- What would happen if we used a different number of intervals (e.g., )?
- Can you explain the choice of as the function in this code?
Tip: When adjusting the number of subintervals in numerical methods like Riemann sums, smaller subintervals (higher ) generally yield more accurate approximations.
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Math Problem Analysis
Mathematical Concepts
Numerical Integration
Riemann Sum
Midpoint Approximation
Exponentials
Formulas
Midpoint Riemann Sum M_n = sum(f(x_mid_i) * delta_x) for i = 0 to n - 1
delta_x = (b - a) / n
f(x) = e^{-x^3}
Theorems
Fundamental Theorem of Calculus (applied in approximating integrals)
Suitable Grade Level
College-level Calculus or Advanced High School Calculus
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