To estimate the area under the graph of f(x) = 1/x from x = 1 to x = 2 using four rectangles and left endpoints, we first need to determine the width of each rectangle.
1. **Calculate the Width of Each Rectangle:**
The total width of the interval from x = 1 to x = 2 is 1 (2 – 1 = 1). We will divide this interval into four equal parts.
Width of each rectangle, Δx = (2 – 1) / 4 = 0.25.
2. **Identify the Left Endpoints:**
The left endpoints for the four rectangles will be:
– x₀ = 1
– x₁ = 1.25
– x₂ = 1.5
– x₃ = 1.75
3. **Calculate the Height of Each Rectangle:**
We will evaluate the function f(x) = 1/x at each left endpoint:
– f(1) = 1/1 = 1
– f(1.25) = 1/1.25 = 0.8
– f(1.5) = 1/1.5 ≈ 0.6667
– f(1.75) = 1/1.75 ≈ 0.5714
4. **Calculate the Area of Each Rectangle:**
The area of each rectangle is given by the formula: Area = Height × Width
– Area₀ = f(1) × Δx = 1 × 0.25 = 0.25
– Area₁ = f(1.25) × Δx = 0.8 × 0.25 = 0.20
– Area₂ = f(1.5) × Δx = 0.6667 × 0.25 ≈ 0.1667
– Area₃ = f(1.75) × Δx = 0.5714 × 0.25 ≈ 0.1429
5. **Estimate the Total Area:**
Now, we can sum the areas of the rectangles to get the total estimated area under the curve:
Total Area ≈ Area₀ + Area₁ + Area₂ + Area₃
Total Area ≈ 0.25 + 0.20 + 0.1667 + 0.1429 ≈ 0.7596.
6. **Sketch the Graph and Rectangles:**
To sketch the graph, plot the curve of f(x) = 1/x from x = 1 to x = 2. Draw each rectangle using the left endpoints as the bases of the rectangles. The height of each rectangle corresponds to the value of f(x) at the left endpoint. This visual representation will help you see how well the rectangles approximate the area under the curve.
This method of using rectangles and left endpoints provides a simple way to estimate the area under a curve, and it’s a foundational concept in calculus.