Final answer:
To approximate the integral ∫¹⁰ √x²+5 dx using the Midpoint Rule with n=4, divide the interval into subintervals, calculate the width and midpoint of each subinterval, evaluate the function at each midpoint, and use the Midpoint Rule formula to approximate the integral.
Step-by-step explanation:
To approximate the integral ∫¹⁰ √x²+5 dx using the Midpoint Rule, divide the interval [1, 10] into 4 subintervals. Calculate the width of each subinterval by subtracting the lower endpoint from the upper endpoint and dividing by n: Δx = (10 - 1) / 4 = 2.25. Find the midpoint of each subinterval by adding the lower endpoint to half of the width: x_i = 1 + (i - 1/2)Δx. Then, evaluate the function at each midpoint value: f(x_i) = √(x_i² + 5). Finally, use the Midpoint Rule formula to approximate the integral: ∫₂M₄ ≈ Δx * (f(x₁) + f(x₂) + f(x₃) + f(x₄)). Round the result to the nearest four decimal places.