Problem
Calculate $M_{6}$ for $f(x)=\sqrt{3 x}$ over $[2,5]$. (Use decimal notation. Give your answer to two deci \[ M_{6} \approx \]
Solution
Step 1 ：Define the function \(f(x) = \sqrt{3x}\).
Step 2 ：Define the interval as \([2, 5]\).
Step 3 ：Set the number of subintervals, \(n\), to 6.
Step 4 ：Calculate the width of each subinterval, \(dx\), using the formula \((b - a) / n\), which gives \(dx = 0.5\).
Step 5 ：Calculate the midpoints of each subinterval. The midpoints are \([2.25, 2.75, 3.25, 3.75, 4.25, 4.75]\).
Step 6 ：Calculate the area of each rectangle by multiplying the function value at the midpoint by the width of the subinterval, and sum them up to get the sixth midpoint Riemann sum, \(M_6\).
Step 7 ：The sixth midpoint Riemann sum of the function \(f(x)=\sqrt{3x}\) over the interval \([2,5]\) is approximately \(\boxed{9.65}\).
