Problem
Use four rectangles to find an estimate of each type for the area under the graph of $f(x)=\sqrt{2 x}$ from $x=0$ to $x=4$. 1. Take the sample points from the left-endpoints. Answer: $L_{4}=$ 2. Is your estimate $L_{4}$ an underestimate or overestimate of the true area? Choose one 3. Take the sample points from the right-endpoints. Answer: $R_{4}=$ 4. Is your estimate $R_{4}$ an underestimate or overestimate of the true area? Choose one 5. Take the sample points from the midpoints. Answer: $M_{4}=$
Solution
Step 1 ：Divide the interval [0,4] into four equal subintervals with a width of \( \frac{4-0}{4}=1 \)
Step 2 ：Calculate the function value at the left-endpoints \( x=0, x=1, x=2, x=3 \) using \( f(x)=\sqrt{2x} \)
Step 3 ：Find the area of each rectangle by multiplying the function value by the width
Step 4 ：Sum up the areas of all four rectangles to get \( L_{4} \)
Step 5 ：\( L_{4}=\boxed{5.863703305156273} \)
