Estimate L4 and R4 over [0,8] for the function f(x)=2squareroot of x. Use decimal notiation. giver your answer to three decimal places.

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.

Estimate L4 and R4 over [0,8] for the function f(x)=2 square root of x. Use decimal notation. Give your answer to three decimal places.

Learn how to estimate the left and right Riemann sums, L4 and R4, for the function f(x) = 2√x over the interval [0, 8]. This problem involves dividing the interval into 4 subintervals and calculating the corresponding sums using decimal notation. Suitable for advanced high school math students in Grades 11-12.

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