Question

Find the approximations T₁₀′​M₁₀′​ and S₁₀​ for ∫π₀​34sin(x)dx. (Round your answers to six decimal places.) T₁₀​= M₁₀​= S₁₀​=​ Find the corresponding errors Eₜ′​Eₘ′​ and Eₛ​. (Round your answers to six decimal places.) Eₜ​= Eₘ​= Eₛ​=​ error bound for Simpson's rule.(Round your answers to six decimal places.) ∣Eₜ​∣≤ ∣Eₘ​∣≤ ∣Eₛ​∣≤​ (c) Using the values of K from part (b), how large do we have to choose n so that the approximations Tₙ′​Mₙ​, and Sₙ​ to the integral in part (a) are accurate to within 0.00001 ? ForTₙn= For Mₙn= For Sₙn=

Answer 1

The student is tasked with approximating a definite integral of a sine function and calculating errors and error bounds for various numerical methods, such as Trapezoidal, Midpoint, and Simpson's Rules.

Step-by-step explanation:

The question is about finding approximations for the definite integral of sine function using different numerical methods and determining the corresponding errors and error bounds for each method. We need to approximate ∫π₀​34sin(x)dx using the Trapezoidal Rule (T₁₀), Midpoint Rule (M₁₀), and Simpson's Rule (S₁₀), and calculate the corresponding errors Eₜ, Eₘ, and Eₛ.

Next, we would compute the error bounds for each rule and determine what value of 'n' is necessary to achieve a given accuracy for each approximation method.