To approximate the area under this curve from $x = 3$ to $x = 7$ using a Left Endpoint approximation with 4 subdivisions, let’s go through the process step-by-step.

Step 1: Define the Interval and Width of Each Subdivision

1. We are given the interval $[3, 7]$.
2. Dividing this interval into 4 equal subdivisions gives each subdivision a width: $\Delta x = \frac{7 - 3}{4} = 1$

Step 2: Determine the Left Endpoints

Since we are using a Left Endpoint approximation, we’ll use the left endpoints of each interval to approximate the area under the curve. The left endpoints in this case are:

• $x = 3$
• $x = 4$
• $x = 5$
• $x = 6$

Step 3: Estimate the Function Values at Left Endpoints

From the graph, we approximate the values of the function at these points:

• $f(3) \approx 1.7$
• $f(4) \approx 2.1$
• $f(5) \approx 2.5$
• $f(6) \approx 2.8$

Step 4: Apply the Left Endpoint Approximation Formula

Using the Left Endpoint approximation, the area $A$ is given by: $A \approx \sum_{i=1}^{4} f(x_i) \cdot \Delta x$ Substituting the values we found: $A \approx (f(3) + f(4) + f(5) + f(6)) \cdot 1$ $A \approx (1.7 + 2.1 + 2.5 + 2.8) \cdot 1$ $A \approx 9.1$

Final Answer

The approximate area under the curve from $x = 3$ to $x = 7$ using a Left Endpoint approximation with 4 subdivisions is: $\boxed{9.1}$

Would you like a deeper explanation or have any questions on the process?

Here are some related questions to explore further:

1. How would a Right Endpoint approximation affect the result?
2. What is the difference between Left Endpoint and Midpoint approximations?
3. How would increasing the number of subdivisions affect the accuracy of the approximation?
4. What is the exact integral of this function over $[3, 7]$ if the function was known?
5. How can we use the Trapezoidal Rule for a similar approximation?

Tip: Using more subdivisions generally increases the accuracy of a Riemann sum approximation because it better captures the curve's behavior.