Question

Approximate the area between the x-axis and f(x) = 2/x from x = 0.5 to x = 2 using a left Riemann sum with 3 equal subdivisions.

Answer 1

The area under the curve f(x) = 2/x from x = 0.5 to x = 2 is approximated using a left Riemann sum with 3 subdivisions. The total area is calculated by summing the areas of the rectangles, resulting in approximately 3.67.

Step-by-step explanation:

To approximate the area between the x-axis and f(x) = 2/x from x = 0.5 to x = 2 using a left Riemann sum with 3 equal subdivisions, we must first determine the width of each subdivision and then calculate the area of each rectangle formed by using the left endpoints to evaluate the function.

The width of each subdivision (Δx) is the difference between the bounds (2 - 0.5) divided by the number of subdivisions (3).

• Δx = (2 - 0.5) / 3 = 0.5

The subdivisions occur at:

• x0 = 0.5
• x1 = 1.0
• x2 = 1.5

We evaluate f(x) at the left endpoints to find the heights:

• f(0.5) = 2/0.5 = 4
• f(1.0) = 2/1.0 = 2
• f(1.5) = 2/1.5 ≈ 1.33

Now we calculate the area of each rectangle formed:

• A1 = Δx × f(0.5) = 0.5 × 4 = 2
• A2 = Δx × f(1.0) = 0.5 × 2 = 1
• A3 = Δx × f(1.5) = 0.5 × 1.33 ≈ 0.67

Finally, the approximation of the total area is the sum of these areas:

• Area ≈ A1 + A2 + A3 = 2 + 1 + 0.67 ≈ 3.67