Questions: Approximate the area under the curve graphed below from x=2 to x=5 using a Left Endpoint approximation with 3 subdivisions.

Solution Steps

The interval from \( x = 2 \) to \( x = 5 \) is divided into 3 subdivisions. The width (\(\Delta x\)) of each subdivision is calculated as follows: \[ \Delta x = \frac{5 - 2}{3} = 1 \]

The left endpoints of the 3 subdivisions are:

• For the first subdivision: \( x = 2 \)
• For the second subdivision: \( x = 3 \)
• For the third subdivision: \( x = 4 \)

Using the graph, approximate the function values at the left endpoints:

• \( f(2) \approx 2 \)
• \( f(3) \approx 3 \)
• \( f(4) \approx 4 \)

The area of each rectangle is given by the function value at the left endpoint times the width (\(\Delta x\)):

• Area of the first rectangle: \( f(2) \cdot \Delta x = 2 \cdot 1 = 2 \)
• Area of the second rectangle: \( f(3) \cdot \Delta x = 3 \cdot 1 = 3 \)
• Area of the third rectangle: \( f(4) \cdot \Delta x = 4 \cdot 1 = 4 \)

Add the areas of the three rectangles to approximate the total area under the curve: \[ \text{Total Area} = 2 + 3 + 4 = 9 \]

Final Answer