To estimate the integrals $L_4$ (Left Riemann Sum) and $R_4$ (Right Riemann Sum) over the interval $[0, 8]$ for the function $f(x) = 2\sqrt{x}$, follow these steps:

Step 1: Calculate the Width of Each Subinterval

The interval $[0, 8]$ is divided into 4 subintervals, so each subinterval has a width of: $\Delta x = \frac{8 - 0}{4} = 2$

Step 2: Determine the Left and Right Endpoints

• Left endpoints for $L_4$: $x_0 = 0$, $x_1 = 2$, $x_2 = 4$, $x_3 = 6$
• Right endpoints for $R_4$: $x_1 = 2$, $x_2 = 4$, $x_3 = 6$, $x_4 = 8$

Step 3: Calculate the Function Values at the Endpoints

Compute $f(x) = 2\sqrt{x}$ at each endpoint.

For $L_4$:

• $f(0) = 2\sqrt{0} = 0.000$
• $f(2) = 2\sqrt{2} = 2.828$
• $f(4) = 2\sqrt{4} = 4.000$
• $f(6) = 2\sqrt{6} = 4.899$

For $R_4$:

• $f(2) = 2\sqrt{2} = 2.828$
• $f(4) = 2\sqrt{4} = 4.000$
• $f(6) = 2\sqrt{6} = 4.899$
• $f(8) = 2\sqrt{8} = 5.657$

Step 4: Calculate $L_4$ and $R_4$

Now sum up the function values for each method and multiply by $\Delta x = 2$.

For $L_4$: $L_4 = \Delta x \times \left( f(0) + f(2) + f(4) + f(6) \right)$ $L_4 = 2 \times (0 + 2.828 + 4.000 + 4.899) = 2 \times 11.727 = 23.454$

For $R_4$: $R_4 = \Delta x \times \left( f(2) + f(4) + f(6) + f(8) \right)$ $R_4 = 2 \times (2.828 + 4.000 + 4.899 + 5.657) = 2 \times 17.384 = 34.768$

Final Answers

• $L_4 = 23.454$
• $R_4 = 34.768$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you calculate the midpoint Riemann sum?
2. How do you determine the exact area under the curve using definite integrals?
3. How do the left and right Riemann sums compare with the exact value of the integral?
4. How would the Riemann sum change if we increased the number of subintervals?
5. What is the trapezoidal rule, and how does it relate to Riemann sums?

Tip: Increasing the number of subintervals in a Riemann sum typically results in a more accurate approximation of the integral.