To find the left-end approximation $L_n$ of the area under the curve $y = f(x)$ using the Riemann sum with the given partition, we can follow these steps:

Step 1: Define the Partition

The partition points given are $\{-7, -4, 1, 3, 7\}$. This divides the interval $[-7, 7]$ into four subintervals:

• $[-7, -4]$
• $[-4, 1]$
• $[1, 3]$
• $[3, 7]$

Step 2: Determine the Widths of the Subintervals

The width ($\Delta x$) of each subinterval is calculated as:

• For $[-7, -4]$: $\Delta x_1 = -4 - (-7) = 3$
• For $[-4, 1]$: $\Delta x_2 = 1 - (-4) = 5$
• For $[1, 3]$: $\Delta x_3 = 3 - 1 = 2$
• For $[3, 7]$: $\Delta x_4 = 7 - 3 = 4$

Step 3: Evaluate the Function at the Left End of Each Subinterval

We need to approximate the value of the function $f(x)$ at the left endpoint of each subinterval:

• $f(-7)$ (left end of $[-7, -4]$)
• $f(-4)$ (left end of $[-4, 1]$)
• $f(1)$ (left end of $[1, 3]$)
• $f(3)$ (left end of $[3, 7]$)

Step 4: Calculate the Left Riemann Sum

The left Riemann sum $L_n$ is given by: $L_n = f(-7) \cdot \Delta x_1 + f(-4) \cdot \Delta x_2 + f(1) \cdot \Delta x_3 + f(3) \cdot \Delta x_4$ Now let's evaluate the function at these points by estimating from the graph.

Approximations from the Graph:

• $f(-7)$ appears to be around 0.5
• $f(-4)$ appears to be around 0.3
• $f(1)$ appears to be around -0.2
• $f(3)$ appears to be around -0.1

Step 5: Compute $L_n$

Substituting the approximate values and widths into the Riemann sum formula:

$L_n = (0.5) \cdot 3 + (0.3) \cdot 5 + (-0.2) \cdot 2 + (-0.1) \cdot 4$

$L_n = 1.5 + 1.5 - 0.4 - 0.4 = 2.2$

Final Answer

The left-end approximation $L_n$ of the area under the curve is approximately 2.2.

Would you like more details or have any other questions?

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Related Questions:

1. What is the interpretation of the left-end Riemann sum in terms of area?
2. How would the Riemann sum change if we used right-end or midpoint approximation?
3. What happens to the approximation as the partition becomes finer?
4. Can you estimate the area using the trapezoidal rule instead of the left Riemann sum?
5. How does the accuracy of a Riemann sum depend on the shape of the function graph?
6. How would you compute the exact area under the curve using integration?
7. What if the function $f(x)$ had points of discontinuity? How would that affect the sum?
8. How do different methods of approximation (left, right, midpoint) compare for a specific function?

Tip: When working with Riemann sums, always visualize the partition and the points being used to make sure you understand which values contribute to the sum.