To approximate the area under the curve from $x = 2$ to $x = 5$ using a Left Endpoint approximation with 3 subdivisions, let's follow these steps:

Step 1: Determine the Width of Each Subdivision

Since we are going from $x = 2$ to $x = 5$ and using 3 subdivisions, the width ($\Delta x$) of each rectangle is: $\Delta x = \frac{5 - 2}{3} = 1$

Step 2: Identify the Left Endpoints

For a Left Endpoint approximation, we use the left end of each interval to find the height of each rectangle. The intervals are:

• $[2, 3]$
• $[3, 4]$
• $[4, 5]$

So, the left endpoints are $x = 2$, $x = 3$, and $x = 4$.

Step 3: Find the Function Values at Each Left Endpoint

Looking at the graph, we approximate the function values at each left endpoint:

1. At $x = 2$, let's assume $f(2) \approx 1.5$
2. At $x = 3$, let's assume $f(3) \approx 2.5$
3. At $x = 4$, let's assume $f(4) \approx 4$

Step 4: Calculate the Area of Each Rectangle

The area of each rectangle is given by $\text{height} \times \text{width}$, where the height is the function value at the left endpoint and the width is $\Delta x = 1$.

1. Area of the rectangle at $x = 2$: $1.5 \times 1 = 1.5$
2. Area of the rectangle at $x = 3$: $2.5 \times 1 = 2.5$
3. Area of the rectangle at $x = 4$: $4 \times 1 = 4$

Step 5: Add the Areas

The total approximate area under the curve is: $1.5 + 2.5 + 4 = 8$

Answer

The approximate area under the curve from $x = 2$ to $x = 5$ using a Left Endpoint approximation with 3 subdivisions is 8 square units.

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:

1. How would the result change if we used a Right Endpoint approximation instead of a Left Endpoint approximation?
2. What if we increased the number of subdivisions to improve the approximation—how would that affect the result?
3. Can we estimate the area under the curve using the Midpoint Rule instead, and how does it compare?
4. How would the exact area compare if we calculated the integral of this function from $x = 2$ to $x = 5$?
5. How does the choice of approximation method (Left, Right, Midpoint) affect accuracy in general?

Tip: As the number of subdivisions increases, Left and Right Endpoint approximations become more accurate, approaching the true area under the curve.