Math Problem Statement

selesaikan masalah ini dengan python: (a)

Tentukan nilai integral berikut dengan menggunakan metode Simpson 1/3 dengan $N = 100$.

\begin{align*}

 \int_{-\pi/2}^{\pi/4} e^{3x}\sin(2x) dx.

\end{align*}

(b)

The integral problem is given as follows

\begin{equation}

    \int_{a}^{b} f(x) dx.

\end{equation}

Then, you will solve the integral problem using the Simpson's 1/3 method and determine the error resulting from this method. The truncation error of Simpson's 1/3 method is defined as

\begin{equation}

    E = \dfrac{(b - a)h^4}{180}f^{(4)}(c), \quad a < c < b.

\end{equation}

Complete the following program to obtain the error resulting from the Simpson's 1/3 method.


import math

import numpy as np

import sympy as sp

from sympy import *




def derivative(f, n):

  if n == 0:

    return _____

  elif n == 1:

    return diff(_____, _____)

  else:

    return diff(derivative(_____, _____), x)




def error_simpson(a, b, c, f, n):

  x = Symbol("x")

  f_prime = derivative(_____, _____)

  fprime = lambdify(_____, _____, 'numpy')

  h = _____

  E = _____

  return E

Solution

import numpy as np import sympy as sp from sympy import Symbol, diff, lambdify, exp, sin, pi

Part (a): Simpson's 1/3 Method for Integration

def simpsons_one_third(f, a, b, N): x = np.linspace(a, b, N+1) h = (b - a) / N integral = f(x[0]) + f(x[-1]) + 4 * np.sum(f(x[1:-1:2])) + 2 * np.sum(f(x[2:-1:2])) integral *= h / 3 return integral

Define the function for part (a)

f_a = lambda x: np.exp(3x) * np.sin(2x) a_a = -np.pi/2 b_a = np.pi/4 N_a = 100

integral_a = simpsons_one_third(f_a, a_a, b_a, N_a)

Part (b): Truncation Error in Simpson's 1/3 Method

def derivative(f, n, x): if n == 0: return f elif n == 1: return diff(f, x) else: return diff(derivative(f, n-1, x), x)

def error_simpson(a, b, f_sympy, n): x = Symbol("x") f_prime = derivative(f_sympy, n, x) f_prime_func = lambdify(x, f_prime, 'numpy') h = (b - a) / 2 c = (a + b) / 2 # Approximate midpoint for worst-case error error = ((b - a) * h**4 / 180) * f_prime_func(c) return error

Define the symbolic function and parameters for part (b)

x = Symbol("x") f_b = sp.exp(3x) * sp.sin(2x) a_b = -np.pi/2 b_b = np.pi/4

Calculate the 4th derivative and truncation error

n_b = 4 error_b = error_simpson(a_b, b_b, f_b, n_b)

integral_a, error_b(2.4334002727173574, 1.31191278099694)(2.4334002727173574, 1.31191278099694)success### Results:

(a) The value of the integral π/2π/4e3xsin(2x)dx\int_{-\pi/2}^{\pi/4} e^{3x}\sin(2x) \, dx using Simpson's 1/3 method with N=100N = 100 is approximately 2.4334.

(b) The truncation error for the Simpson's 1/3 method in this case is approximately 1.3119.

Would you like a detailed explanation of the process or the Python code? Here are some questions to consider for deeper understanding:

  1. What is the significance of choosing N=100N = 100 for Simpson's 1/3 method?
  2. How does the truncation error formula relate to the choice of NN in numerical integration?
  3. What are the key advantages of using Simpson's 1/3 rule over other numerical methods?
  4. Could this method be extended to functions with singularities or discontinuities?
  5. How does the fourth derivative affect the magnitude of the truncation error?

Tip: Increasing NN in Simpson's 1/3 method reduces the step size hh, which in turn decreases the truncation error.

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Math Problem Analysis

Mathematical Concepts

Numerical Integration
Simpson's 1/3 Method
Error Analysis
Exponential and Trigonometric Functions

Formulas

Simpson's 1/3 Rule: ∫(a)^(b) f(x) dx ≈ (h/3) [f(a) + 4f(a+h) + 2f(a+2h) + ... + f(b)]
Truncation Error: E = ((b - a)h^4 / 180)f^(4)(c), a < c < b

Theorems

Simpson's Rule
Error Bound for Numerical Integration

Suitable Grade Level

University Level