https://file.aiquickdraw.com/mathbot/file/173050724305015uq017u.jpg

Approximate the area under the curve graphed below from x=3 
 to x=7 
 using a Left Endpoint approximation with 4 subdivisions.

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?

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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.

Approximate the area under the curve graphed below from x=3 to x=7 using a Left Endpoint approximation with 4 subdivisions.

Approximate the area under a curve from x=3 to x=7 using a Left Endpoint Riemann Sum with 4 subdivisions. This problem covers essential calculus concepts like Riemann sums, left endpoint approximations, and step-by-step calculations for area estimation. Suitable for high school students in Grades 11-12.

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