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The problem asks to evaluate the **left** and **right Riemann sums** for \( f \) over the interval \( [0, 2] \) with \( n = 4 \) using the provided table of values.

The given dataThe problem asks to evaluate the **left** and **right Riemann sums** for \( f \) over the interval \( [0, 2] \) with \( n = 4 \) using the provided table of values.

The given 
- Interval: \( [0, 2] \)
- Number of subintervals: \( n = 4 \)
- Values for \( x \) and \( f(x) \):

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 0 & 0.5 & 1 & 1.5 & 2 \\
\hline
f(x) & 5 & 3 & 2 & 1 & 1 \\
\hline
\end{array}
\]

### Step 1: Determine the width of each subinterval
The width \( \Delta x \) of each subinterval is calculated as:
\[
\Delta x = \frac{b - a}{n} = \frac{2 - 0}{4} = 0.5
\]

### Step 2: Left Riemann sum
For the **left Riemann sum**, we use the left endpoint of each subinterval. The left endpoints are \( x = 0, 0.5, 1, 1.5 \), and the corresponding values of \( f(x) \) are 5, 3, 2, and 1.

The left Riemann sum is:
\[
L_4 = \Delta x \left( f(0) + f(0.5) + f(1) + f(1.5) \right)
\]
\[
L_4 = 0.5 \times (5 + 3 + 2 + 1) = 0.5 \times 11 = 5.5
\]

### Step 3: Right Riemann sum
For the **right Riemann sum**, we use the right endpoint of each subinterval. The right endpoints are \( x = 0.5, 1, 1.5, 2 \), and the corresponding values of \( f(x) \) are 3, 2, 1, and 1.

The right Riemann sum is:
\[
R_4 = \Delta x \left( f(0.5) + f(1) + f(1.5) + f(2) \right)
\]
\[
R_4 = 0.5 \times (3 + 2 + 1 + 1) = 0.5 \times 7 = 3.5
\]

### Final Results:
- Left Riemann sum: \( 5.5 \)
- Right Riemann sum: \( 3.5 \)

Would you like further details or clarifications on the process?

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Here are five related questions to explore the concept further:
1. How does increasing the number of subintervals \( n \) affect the accuracy of the Riemann sums?
2. What is the significance of using the left versus right Riemann sums in estimating area under a curve?
3. How would you compute the midpoint Riemann sum for the same data?
4. Can you derive the actual integral of \( f(x) \) over \( [0, 2] \) using a known function for \( f(x) \)?
5. How do Riemann sums relate to the concept of definite integrals?

**Tip**: The more subintervals you use (larger \( n \)), the more accurate your Riemann sum approximation becomes for the actual integral!

Evaluate the left and right Riemann sums for f over the interval [0, 2] with n = 4 using the provided table of values for f(x).

Learn how to compute the left and right Riemann sums for f(x) over the interval [0, 2] with n = 4 using a table of values. This calculus problem demonstrates the steps for numerical approximation of definite integrals using Riemann sums.

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