To calculate $M_6$, the approximation of the integral of $f(x) = \sqrt{7x}$ over $[3,6]$ using the midpoint rule with $n = 6$, follow these steps:

Step 1: Determine the width of each subinterval

$\Delta x = \frac{b-a}{n} = \frac{6-3}{6} = 0.5$

Step 2: Identify the midpoints of the subintervals

The subintervals are: [ [3, 3.5], [3.5, 4], [4, 4.5], [4.5, 5], [5, 5.5], [5.5, 6] ] The midpoints are: $x_1 = 3.25, \, x_2 = 3.75, \, x_3 = 4.25, \, x_4 = 4.75, \, x_5 = 5.25, \, x_6 = 5.75$

Step 3: Evaluate $f(x) = \sqrt{7x}$ at each midpoint

$f(x_1) = \sqrt{7 \cdot 3.25} = \sqrt{22.75} \approx 4.77$ $f(x_2) = \sqrt{7 \cdot 3.75} = \sqrt{26.25} \approx 5.12$ $f(x_3) = \sqrt{7 \cdot 4.25} = \sqrt{29.75} \approx 5.45$ $f(x_4) = \sqrt{7 \cdot 4.75} = \sqrt{33.25} \approx 5.77$ $f(x_5) = \sqrt{7 \cdot 5.25} = \sqrt{36.75} \approx 6.06$ $f(x_6) = \sqrt{7 \cdot 5.75} = \sqrt{40.25} \approx 6.35$

Step 4: Apply the midpoint rule

The formula for the midpoint rule is: $M_6 = \Delta x \sum_{i=1}^6 f(x_i)$ Substitute the values: $M_6 = 0.5 \cdot (4.77 + 5.12 + 5.45 + 5.77 + 6.06 + 6.35)$ $M_6 = 0.5 \cdot 33.52 = 16.76$

Final Answer:

$M_6 \approx 16.76$

Let me know if you have any questions or need further details!

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Follow-up Questions:

1. How is the midpoint rule different from other numerical integration methods like the trapezoidal rule?
2. Why is it important to use decimal notation in approximations?
3. Can $M_6$ be calculated using a smaller or larger $n$? How does $n$ affect accuracy?
4. How would the integral be solved analytically for $\int_3^6 \sqrt{7x} \, dx$?
5. What is the geometric interpretation of the midpoint rule?

Tip: Always double-check intermediate calculations when using numerical methods to avoid compounding errors!