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 using Simpson's 1/3 method with 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:
- What is the significance of choosing for Simpson's 1/3 method?
- How does the truncation error formula relate to the choice of in numerical integration?
- What are the key advantages of using Simpson's 1/3 rule over other numerical methods?
- Could this method be extended to functions with singularities or discontinuities?
- How does the fourth derivative affect the magnitude of the truncation error?
Tip: Increasing in Simpson's 1/3 method reduces the step size , which in turn decreases the truncation error.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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
Related Recommendation
Simpson's Rule Approximation with n=50 and n=100 for Integral of 4sin(x)/x from 0 to π/2
Simpson's Rule Approximation of sin(x) from 0 to π with Error Analysis
Approximating the Integral of x^4*cos(x) Using Simpson's Rule
Solving the Integral of x^3 * e^(2x) * sqrt(3 - x) using Simpson's Rule
Numerical Integration: Approximate Integral of e^(-3x^2) from 0 to 1 Using Trapezoidal, Midpoint, and Simpson's Rules with n=4