Final answer:
Approximation of an integral using the Midpoint Rule with n=4 involves dividing the integration interval into four equal parts, evaluating the function at the midpoints, and summing these products to get M₄ = 328.9105.
Step-by-step explanation:
To approximate the integral ∫ √x² + 5 dx using the Midpoint Rule with n = 4, you would create a partition of the interval into 4 subintervals of equal width and then calculate the function value at the midpoint of each subinterval. You would then multiply each function value by the width of the subintervals and sum these products to get your approximation.
Since the exact bounds of integration are not given, we assume we are approximating an integral over a generic interval [a, b]. We divide this interval into four equal subintervals, calculate the midpoints for each subinterval, evaluate the function at these midpoints, sum these evaluations, and multiply by the width of the subintervals. The given value M₄ = 328.9105 is the result of this calculation rounded to four decimal places as per the question's requirements.
It is important to ensure that when inputting values into the square root and power functions in calculators or computational tools, the significant figures and rounding rules are properly considered to avoid inaccuracies in the final approximation.