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Evaluate the following integral: TT/2 I Jo (8 + 4 cos x) dx a) analytically b) single application of the trapezoidal rule c) multiple-application trapezoidal rule, with n = 2 and 4 d) single applicatio (2 mila

Evaluate the following integral: TT/2 I Jo (8 + 4 cos x) dx a) analytically b) single application of the trapezoidal rule c) multiple-application trapezoidal rule, with n = 2 and 4 d) single applicatio (2 mila

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Problem Evaluate the following integral: π/2 I JO (8 + 4 cos x) dx a) analytically b) single application of the trapezoidal rule c) multiple-application trapezoidal rule, with n = 2 and 4 d) single application of Simpson's 1/3 rule e) multiple-application Simpson's 1/3 rule, with n = 4 f) single application of Simpson's 3/8 rule g) multiple-application Simpson's rule, with n = 5. For each of the numerical estimates (b) through (g), determine the percent relative error based on (a).
Transcribed Image Text:Problem Evaluate the following integral: π/2 I JO (8 + 4 cos x) dx a) analytically b) single application of the trapezoidal rule c) multiple-application trapezoidal rule, with n = 2 and 4 d) single application of Simpson's 1/3 rule e) multiple-application Simpson's 1/3 rule, with n = 4 f) single application of Simpson's 3/8 rule g) multiple-application Simpson's rule, with n = 5. For each of the numerical estimates (b) through (g), determine the percent relative error based on (a).
Problem 2 Use two-, three-, and four-point Gauss-Legendre integration formulas to evaluate 3 √₁²(2x + ²)² dx Use the analytical solution of the integral to determine the true percent relative error Et for each case. Problem 3 Compute the second and third derivatives using central difference approximations of O(h) for each of the following functions at the specified location and for the specified step size a) y = x³ + 4x-15 at x = 0, h = 0.25 b) y=x² cos x at x = 0.4, h = 0.1 c) y = sin(0.5 √x)/x at x = 1, h = 0.2 Compare your results with the analytical solutions.
Transcribed Image Text:Problem 2 Use two-, three-, and four-point Gauss-Legendre integration formulas to evaluate 3 √₁²(2x + ²)² dx Use the analytical solution of the integral to determine the true percent relative error Et for each case. Problem 3 Compute the second and third derivatives using central difference approximations of O(h) for each of the following functions at the specified location and for the specified step size a) y = x³ + 4x-15 at x = 0, h = 0.25 b) y=x² cos x at x = 0.4, h = 0.1 c) y = sin(0.5 √x)/x at x = 1, h = 0.2 Compare your results with the analytical solutions.
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