as np

Define the function to integrate

def f(x): return np.sqrt(x)

Define the limits of integration and number of subintervals

a = 5 # Lower limit b = 10 # Upper limit n = 5 # Number of subintervals

Calculate the width of each subinterval

dx = (b - a) / n

Right endpoints

right_endpoints = [a + i * dx for i in range(1, n + 1)] right_sum = sum(f(x) * dx for x in right_endpoints)

Left endpoints

left_endpoints = [a + i * dx for i in range(n)] left_sum = sum(f(x) * dx for x in left_endpoints)

Midpoints

midpoints = [a + (i + 0.5) * dx for i in range(n)] midpoint_sum = sum(f(x) * dx for x in midpoints)

right_sum_rounded = round(right_sum, 2) left_sum_rounded = round(left_sum, 2) midpoint_sum_rounded = round(midpoint_sum, 2)

(right_sum_rounded, left_sum_rounded, midpoint_sum_roundedHere are the approximate values of the integral $\int_{5}^{10} \sqrt{x} \, dx$ using Riemann sums with 5 subintervals:

a) Using right endpoints: $14.09$
b) Using left endpoints: $13.16$
c) Using midpoints: $13.63$

Do you want detailed steps on how these were calculated or have any specific questions?

Here are some follow-up questions:

1. How is the width of each subinterval calculated?
2. Why does the choice of endpoints (right, left, or midpoints) affect the result?
3. What does the approximation indicate about the behavior of $\sqrt{x}$ on $[5, 10]$?
4. Can Riemann sums provide an exact value for all integrals?
5. How could increasing the number of subintervals improve the accuracy?

Tip: When approximating integrals, midpoints usually provide a more accurate estimate compared to using left or right endpoints.