Apply the composite Trapezoidal, Midpoint and Simpson's rules to approximate the integral f(x)dx for a = 0, b = 1, f(x) = x² + 3x using N = 10 rectangles. Calculate the absolute error using the exact integral Repeat your calculation for N = 10, 20, 30, 40, 50. Discuss your observation for each method and compare the perfomance of all three methods. Program Flow • Import necessary Python libraries • Define the integrand f(x) • Define a function for the single trapezoidal rule given the input variables ▪ left end point of interval ▪ right end pont of interval ▪ integrand f • Determine the length of each subinterval h • Construct a partition of the integration domain [a, b] • For each subinterval ▪ calculate the approximate integral on subinteval [x;-1-x;] ▪ add the result to all previous subintervals • Calculate the absolute error • Display the final approximate integral and the absolute error
Apply the composite Trapezoidal, Midpoint and Simpson's rules to approximate the integral f(x)dx for a = 0, b = 1, f(x) = x² + 3x using N = 10 rectangles. Calculate the absolute error using the exact integral Repeat your calculation for N = 10, 20, 30, 40, 50. Discuss your observation for each method and compare the perfomance of all three methods. Program Flow • Import necessary Python libraries • Define the integrand f(x) • Define a function for the single trapezoidal rule given the input variables ▪ left end point of interval ▪ right end pont of interval ▪ integrand f • Determine the length of each subinterval h • Construct a partition of the integration domain [a, b] • For each subinterval ▪ calculate the approximate integral on subinteval [x;-1-x;] ▪ add the result to all previous subintervals • Calculate the absolute error • Display the final approximate integral and the absolute error
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Python
![In [ ]:
In [ ]:
In [ ]:
Example
Apply the composite Trapezoidal, Midpoint and Simpson's rules to approximate the integral f(x)dx for a = 0, b = 1, f(x) = x² + 3x using N = 10
rectangles. Calculate the absolute error using the exact integral. Repeat your calculation for N = 10, 20, 30, 40, 50. Discuss your observation for each
method and compare the perfomance of all three methods.
Program Flow
• Import necessary Python libraries
• Define the integrand f(x)
•
Define a function for the single trapezoidal rule given the input variables
■ left end point of interval
▪ right end pont of interval
▪ integrand f
Determine the length of each subinterval h = b/d
N
Construct a partition of the integration domain [a, b]
• For each subinterval
.
▪ calculate the approximate integral on subinteval [x;-1 - x;]
▪ add the result to all previous subintervals
Calculate the absolute error
• Display the final approximate integral and the absolute error
Trapezoidal Rule
Midpoint Rule
Simpson's Rule](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4aac48a-4d20-41e4-aba5-4ec9a336b2fd%2F73b07a3e-da4a-41b5-b0ee-7c27b6ab748c%2F9sm1mi_processed.png&w=3840&q=75)
Transcribed Image Text:In [ ]:
In [ ]:
In [ ]:
Example
Apply the composite Trapezoidal, Midpoint and Simpson's rules to approximate the integral f(x)dx for a = 0, b = 1, f(x) = x² + 3x using N = 10
rectangles. Calculate the absolute error using the exact integral. Repeat your calculation for N = 10, 20, 30, 40, 50. Discuss your observation for each
method and compare the perfomance of all three methods.
Program Flow
• Import necessary Python libraries
• Define the integrand f(x)
•
Define a function for the single trapezoidal rule given the input variables
■ left end point of interval
▪ right end pont of interval
▪ integrand f
Determine the length of each subinterval h = b/d
N
Construct a partition of the integration domain [a, b]
• For each subinterval
.
▪ calculate the approximate integral on subinteval [x;-1 - x;]
▪ add the result to all previous subintervals
Calculate the absolute error
• Display the final approximate integral and the absolute error
Trapezoidal Rule
Midpoint Rule
Simpson's Rule
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