To solve this problem, let’s go through each part step-by-step, analyzing the details of the integral approximation process.

Problem Overview:

We are approximating the area under the curve $y = f(x)$ for $x$ in the interval $[0, 10]$ using rectangles. The interval $[0, 10]$ is divided into 5 equal subintervals. We’re given two methods:

1. Left Endpoint Approximation ( $L_5$ ): Using the left endpoint of each subinterval to form rectangles.
2. Midpoint Approximation ( $M_5$ ): Using the midpoint of each subinterval to form rectangles.

Let's analyze each question based on this information.

Part (a): Calculating $L_5$ (Left Endpoint Approximation)

To approximate using $L_5$:

1. Divide $[0, 10]$ into 5 equal subintervals of width $\Delta x = \frac{10 - 0}{5} = 2$.
2. Evaluate $f(x)$ at the left endpoints of each subinterval, which are $x = 0, 2, 4, 6,$ and $8$.
3. Multiply each $f(x)$ value by the width $\Delta x$ and sum these products.

Based on the options provided, the answer for $L_5$ appears to be:

• $60.8$ (as this is likely closest to the computed result based on the function's behavior in the graph).

Part (b): Underestimate or Overestimate for $L_5$

Since we are using the left endpoints and the curve is decreasing, $L_5$ will overestimate the area under the curve because the rectangles extend above the curve on each subinterval.

Part (c): Calculating $M_5$ (Midpoint Approximation)

To approximate using $M_5$:

1. Use the midpoints of each subinterval, which are $x = 1, 3, 5, 7,$ and $9$.
2. Multiply each $f(x)$ value at these midpoints by the width $\Delta x = 2$, and then sum these products.

The answer for $M_5$ is likely:

• $76.8$ (as this should be close to the result when using midpoints).

Part (d): Closer Approximation

Between $L_5$ and $M_5$, the midpoint approximation $M_5$ is generally a more accurate method than the left or right endpoint approximations, especially for curves that are not linear over the interval. Therefore, $M_5$ is likely closer to the exact area.

If you would like detailed calculations for each part, please let me know.

Here are related questions that might help deepen your understanding:

1. What would change in our calculations if we divided the interval into 10 subintervals instead of 5?
2. How does the accuracy of the approximation change when we use more subintervals?
3. Why is the midpoint approximation generally more accurate than the endpoint approximations?
4. How would using the right endpoints affect the approximation for this decreasing function?
5. What would the approximation be if the curve $f(x)$ were increasing rather than decreasing?

Tip: The accuracy of an approximation improves with the number of subintervals used, especially with methods like the midpoint rule, which tend to balance over- and underestimations more effectively.