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

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

Solution

Solution Steps

Step 1: Determine the width of each subdivision

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

Step 2: Identify the left endpoints

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 $

Step 3: Evaluate the function at the left endpoints

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

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

Step 4: Calculate the area of each rectangle

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 $

Step 5: Sum the areas of the rectangles

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

Final Answer

The approximate area under the curve from $ x = 2 $ to $ x = 5 $ using a Left Endpoint approximation with 3 subdivisions is $ 9 $.

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