Final answer:
a) T₁₀ ≈ 45.612967, M₁₀ ≈ 45.450654
b) |Eₜ|≤ 0.013248, |Eₘ|≤ 0.150934
c) n = 43 for Tₙ, n = 6 for Mₙ
Step-by-step explanation:
To approximate the integral ∫₁² 39e¹/ˣ dx, both the Trapezoidal Rule (T) and Midpoint Rule (M) approximations are utilized with 10 subintervals (n = 10) in this interval. Applying the Trapezoidal Rule (T), T₁₀, and the Midpoint Rule (M), M₁₀, the approximate values are found to be T₁₀ ≈ 45.612967 and M₁₀ ≈ 45.450654 when rounded to six decimal places.
Next, to estimate the errors for both approximations, the absolute values of the differences between the actual integral and the approximations are computed. |Eₜ| and |Eₘ| represent the maximum errors for the Trapezoidal Rule and Midpoint Rule, respectively. For this integral, |Eₜ| is estimated to be ≤ 0.013248 and |Eₘ| is ≤ 0.150934, rounded to six decimal places.
To achieve an accuracy within 0.0001 for both approximations, calculations for the number of subintervals (n) required are performed. For the Trapezoidal Rule (Tₙ), the error bound formula is applied to find the appropriate n, which results in n = 43. For the Midpoint Rule (Mₙ), n = 6 subintervals suffice to achieve the required accuracy of 0.0001. These values of n ensure that the approximations Tₙ and Mₙ are accurate within the specified limit.
Understanding the Trapezoidal Rule and Midpoint Rule along with their respective error estimations provides a method to determine the number of subintervals necessary to achieve a desired accuracy in approximating definite integrals. These rules are invaluable tools in numerical analysis, enabling estimation of integrals without evaluating complex functions directly. Adjusting the number of subintervals can refine the accuracy of the approximations to meet specific precision requirements for various applications.