Calculating the Area Bounded by xy 1, x 0, y 1, and y 2

In this article, we will explore how to find the area bounded by the curves xy 1, x 0, y 1, and y 2. This problem involves concepts from integral calculus and the use of natural logarithms. We will guide you through the process step-by-step.

Identifying the Curves and Their Intersections

The problem involves four distinct curves and lines. Let's identify them:

xy 1: This is a hyperbola with branches in the first and third quadrants, represented by the function y 1/x. x 0: This is the y-axis, a vertical line passing through the origin. y 1: This is a horizontal line passing through the point (0, 1). y 2: This is a horizontal line passing through the point (0, 2).

Finding Points of Intersection

To find where the curves and lines intersect:

Intersection of y 1/x and y 1:

Setting y 1 in the equation y 1/x:

1 1/x implies x 1. Therefore, the intersection point is (1, 1).

Intersection of y 1/x and y 2:

Setting y 2 in the equation y 1/x:

2 1/x implies x 1/2. Therefore, the intersection point is (1/2, 2).

Setting Up the Integral for the Area

The area we want to find is bounded between the horizontal lines y 1 and y 2 under the curve y 1/x. We need to integrate this function from x 1/2 to x 1.

The area A can be expressed as:

A ∫1/21(1/x - 1)dx ∫12(1/x - 2)dx

Calculating the First Integral

Let's calculate the first part of the area A_1 using the limits x 1/2 to x 1.

A_1 ∫1/21(1/x - 1)dx ∫1/21(1/x)dx - ∫1/211dx

First part: ∫(1/x)dx ln(x) yielding: [ln(x)]1/21 ln(1) - ln(1/2) 0 - ln(1/2) ln(2) Second part: ∫1dx x yielding: [x]1/21 1 - 1/2 1/2

So, A_1 ln(2) - 1/2.

Calculating the Second Integral

Now let's calculate the second part of the area A_2 using the limits x 1 to x 2.

A_2 ∫12(1/x - 2)dx ∫12(1/x)dx - ∫122dx

First part: ∫(1/x)dx ln(x) yielding: [ln(x)]12 ln(2) - ln(1) ln(2) Second part: ∫2dx 2x yielding: [2x]12 4 - 2 2

So, A_2 ln(2) - 2.

Combining the Areas

The total area A is the sum of A_1 and A_2 as follows:

A A_1 A_2 (ln(2) - 1/2) (ln(2) - 2) 2ln(2) - 5/2

Therefore, the total area bounded by the curves is:

boxed{2ln(2) - 5/2}

Brief Summary

This problem demonstrates the application of integral calculus to find the area bounded by specific curves and lines. By identifying the curves and their points of intersection, setting up appropriate integrals, and carefully evaluating the limits, we can calculate the exact area.