Introduction

The problem at hand is to determine the area bounded by the line xy 10 and both the coordinate axes. This problem involves understanding the relationship between the line, the coordinate axes, and how to calculate the enclosed area geometrically. In this article, we will explore a straightforward method to solve this problem and verify the result using integration.

Understanding the Line Equation: xy 10

The equation xy 10 describes a hyperbola rather than a straight line. However, for the purpose of this problem, we will consider the intersections with the x and y axes and then determine the enclosed area by the triangle formed.

Intersection Points and Triangle Formed

To find the intersection points with the coordinate axes, we can substitute 0 for y and x separately into the equation xy 10.

Intersection with the x-axis

When y 0, the equation xy 10 becomes x0 0, which is not a valid solution for x 10. This indicates that the line does not intersect the x-axis at y 0 but rather at x 10 when y 0 (i.e., the point (10, 0)).

Intersection with the y-axis

When x 0, the equation xy 10 becomes 0y 0, which again is not a valid solution for y 0 but rather at y 10 when x 0 (i.e., the point (0, 10)).

These points, (10, 0) and (0, 10), along with the origin (0, 0), form an isosceles right-angled triangle. The base and height of this triangle are both 10 units.

Calculating the Area of the Triangle

The area of a right-angled triangle is given by:

Area (1/2) * base * height

Substituting the values, we have:

Area (1/2) * 10 * 10 50 square units

Verification Using Integration

To verify the result, we can use integration. The line xy 10 intersects the x-axis at x 10 and the y-axis at y 10. We can set up definite integrals to find the area under the curve.

Using Integration for Area Under the Curve

Consider the equation xy 10. Solving for y, we get:

y 10/x

We can now set up the integral to find the area between x 0 and x 10:

Area ∫010 (10/x) dx

Evaluating the integral:

Area 10 * [ln(x)]010

Since the natural logarithm of 0 is undefined, we consider the region from a small positive value close to 0 to 10. This simplifies to:

Area ≈ 10 * [ln(10) - ln(ε)] ≈ 10 * ln(10) ≈ 23.02585

This result is an approximation and not directly applicable to the problem as stated, since the correct intersection points yield a more straightforward geometric solution.

However, if we consider the enclosed triangular region directly, the area calculated geometrically is 50 square units, which is consistent with the earlier geometric derivation.

Conclusion

In summary, the area bounded by the line xy 10 and both the coordinate axes is 50 square units, as determined by the intersection points and the geometric properties of the enclosed triangle.