### Understanding the Intersection of Graphs: Figure 14In this figure, we analyze the intersection and the enclosed region of three different functions plotted on the Cartesian plane. The functions are:1. \( y = x \)2. \( y = \frac{1}{x} \)3. \( y = \frac{1}{4}x \)#### Detailed Explanation of Graphs1. **The Line \( y = x \)**: - This graph is a straight line passing through the origin (0,0) with a slope of 1. It represents a linear relationship where the value of \( y \) is equal to the value of \( x \).2. **The Curve \( y = \frac{1}{x} \)**: - This graph is a hyperbola, which represents a reciprocal function. For positive values of \( x \), the curve lies in the first quadrant and decreases as \( x \) increases.3. **The Line \( y = \frac{1}{4}x \)**: - This graph is another linear function passing through the origin, but it has a smaller slope of 0.25 or \( \frac{1}{4} \), making it less steep compared to the line \( y = x \).#### Intersection Points and Enclosed Region- The three graphs intersect at various points on the Cartesian plane.- The blue shaded region represents the area bounded by these three functions.- The points of intersection are critical in determining the limits of integration if we were to calculate the area of the shaded region.#### Points of Interest and Intersection- The line \( y = x \) intersects \( y = \frac{1}{x} \) above the y-axis.- The line \( y = \frac{1}{4}x \) intersects \( y = \frac{1}{x} \) at a point beyond \( x = 1 \).- These intersection points demarcate the boundaries of the shaded region, highlighting the area where \( y \), determined by each of the three functions, changes from one dominance to another.### ConclusionFigure 14 provides a visual representation of how different mathematical functions interact on a coordinate plane, focusing particularly on their points of intersection and the areas they enclose. Understanding these graphical properties allows us to visualize solutions and areas under curves, which is fundamental in calculus and analytical geometry. **Example 6**Find the area of the region enclosed by the curves \( y = \frac{1}{x} \), \( y = x \), and \( y = \frac{1}{4}x \), using: (a) \( x \) as the variable of integration (b) \( y \) as the variable of integration.**Solution** The region is graphed in Figure 14.[Note to educators: Include the graph from Figure 14 here. The graph should show the curves \( y = \frac{1}{x} \), \( y = x \), and \( y = \frac{1}{4}x \), and the enclosed region whose area is to be calculated.]

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Answered: EXAMPLE 6 Find the area of the region…

EXAMPLE 6 Find the area of the region enclosed by the curves y = 1/x, y = x, and y = x, using (a) x as the variable of integration and (b) y as the variable of integration.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

### Understanding the Intersection of Graphs: Figure 14

In this figure, we analyze the intersection and the enclosed region of three different functions plotted on the Cartesian plane. The functions are:

1. \( y = x \)
2. \( y = \frac{1}{x} \)
3. \( y = \frac{1}{4}x \)

#### Detailed Explanation of Graphs

1. **The Line \( y = x \)**:
- This graph is a straight line passing through the origin (0,0) with a slope of 1. It represents a linear relationship where the value of \( y \) is equal to the value of \( x \).

2. **The Curve \( y = \frac{1}{x} \)**:
- This graph is a hyperbola, which represents a reciprocal function. For positive values of \( x \), the curve lies in the first quadrant and decreases as \( x \) increases.

3. **The Line \( y = \frac{1}{4}x \)**:
- This graph is another linear function passing through the origin, but it has a smaller slope of 0.25 or \( \frac{1}{4} \), making it less steep compared to the line \( y = x \).

#### Intersection Points and Enclosed Region

- The three graphs intersect at various points on the Cartesian plane.
- The blue shaded region represents the area bounded by these three functions.
- The points of intersection are critical in determining the limits of integration if we were to calculate the area of the shaded region.

#### Points of Interest and Intersection
- The line \( y = x \) intersects \( y = \frac{1}{x} \) above the y-axis.
- The line \( y = \frac{1}{4}x \) intersects \( y = \frac{1}{x} \) at a point beyond \( x = 1 \).
- These intersection points demarcate the boundaries of the shaded region, highlighting the area where \( y \), determined by each of the three functions, changes from one dominance to another.

### Conclusion

Figure 14 provides a visual representation of how different mathematical functions interact on a coordinate plane, focusing particularly on their points of intersection and the areas they enclose. Understanding these graphical properties allows us to visualize solutions and areas under curves, which is fundamental in calculus and analytical geometry.

**Example 6**
Find the area of the region enclosed by the curves \( y = \frac{1}{x} \), \( y = x \), and \( y = \frac{1}{4}x \), using:

(a) \( x \) as the variable of integration
(b) \( y \) as the variable of integration.

**Solution**
The region is graphed in Figure 14.

[Note to educators: Include the graph from Figure 14 here. The graph should show the curves \( y = \frac{1}{x} \), \( y = x \), and \( y = \frac{1}{4}x \), and the enclosed region whose area is to be calculated.]

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