Problem 22
Question
Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the triangle with vertices \((1,0),(1,1)\), and \((2,0)\).
Step-by-Step Solution
Verified Answer
Inequalities for \( R \) are: \( 1 \leq x \leq 2 \), \( 0 \leq y \leq -x + 2 \) and \( 0 \leq y \leq 1 \), \( 1 \leq x \leq -y + 2 \).
1Step 1: Identify the boundaries
Determine the equations of the lines that form the boundaries of the triangle. The triangle is formed by the vertices \( (1,0) \), \( (1,1) \), and \( (2,0) \). The lines are vertical line \( x = 1 \), horizontal line \( y = 0 \) and the line connecting \( (1,1) \) and \( (2,0) \).
2Step 2: Find the equation of the diagonal boundary
The line connecting \( (1,1) \) and \( (2,0) \) can be found using the point-slope form: \[ y - 1 = \frac{0 - 1}{2 - 1}(x - 1) \Rightarrow y = -x + 2 \]
3Step 3: Determine vertical cross sections
For vertical cross sections, the x-values range from \( x = 1 \) to \( x = 2 \). For a fixed \( x \) in this interval, the y-values range from \( y = 0 \) to \( y = -x + 2 \).
4Step 4: Determine horizontal cross sections
For horizontal cross sections, the y-values range from \( y = 0 \) to \( y = 1 \). For a fixed \( y \) in this interval, the x-values range from \( x = 1 \) to the line \( x = -y + 2 \).
5Step 5: Set up inequalities for the region
Combining these observations, the region \( R \) can be expressed with the inequalities for vertical cross sections \[ 1 \leq x \leq 2 \quad \text{and} \quad 0 \leq y \leq -x + 2 \] and for horizontal cross sections \[ 0 \leq y \leq 1 \quad \text{and} \quad 1 \leq x \leq -y + 2 \]
Key Concepts
Vertical Cross SectionsHorizontal Cross SectionsTriangle Region InequalitiesBoundary Equations
Vertical Cross Sections
The concept of vertical cross sections in geometry helps us understand the vertical 'slices' or sections of a figure.
In this exercise, vertical cross sections of the triangle region take place along lines of fixed x-values, while y-values change.
Given the vertices \( (1,0), (1,1), (2,0) \), the x-values range from x = 1 to x = 2.
For any vertical line within this range, y-values will span from y = 0 (bottom boundary) up to y = -x + 2 (diagonal boundary).
This concept helps visualize the triangle as layers of vertical slices, each defined by the y-values within the fixed x-range.
In this exercise, vertical cross sections of the triangle region take place along lines of fixed x-values, while y-values change.
Given the vertices \( (1,0), (1,1), (2,0) \), the x-values range from x = 1 to x = 2.
For any vertical line within this range, y-values will span from y = 0 (bottom boundary) up to y = -x + 2 (diagonal boundary).
This concept helps visualize the triangle as layers of vertical slices, each defined by the y-values within the fixed x-range.
Horizontal Cross Sections
Just like vertical cross sections, horizontal cross sections divide a region along horizontal lines of fixed y-values.
For the triangle, y-values range from y = 0 to y = 1.
Within this range, x-values vary between x = 1 (left boundary) and x = -y + 2 (diagonal boundary).
This approach allows us to see the entire triangle as a collection of horizontal slices.
Each slice is characterized by the x-values allowed for the fixed y-value.
This method is particularly useful for understanding the shape and boundaries of irregular figures.
For the triangle, y-values range from y = 0 to y = 1.
Within this range, x-values vary between x = 1 (left boundary) and x = -y + 2 (diagonal boundary).
This approach allows us to see the entire triangle as a collection of horizontal slices.
Each slice is characterized by the x-values allowed for the fixed y-value.
This method is particularly useful for understanding the shape and boundaries of irregular figures.
Triangle Region Inequalities
Inequalities are a powerful tool for describing regions in geometry, particularly triangles.
For our triangle with vertices \( (1,0), (1,1), (2,0) \), we derive inequalities by combining the information from vertical and horizontal cross sections.
Vertically, the inequalities are: \[ 1 \leq x \leq 2 \quad \text{and} \quad 0 \leq y \leq -x \+ \2 \]
Horizontally, they're given by: \[ 0 \leq y \leq 1 \quad \text{and} \quad 1 \leq x \leq -y \+ \2 \]
These inequalities define the region inside the triangle by highlighting all possible x and y values within the boundaries.
For our triangle with vertices \( (1,0), (1,1), (2,0) \), we derive inequalities by combining the information from vertical and horizontal cross sections.
Vertically, the inequalities are: \[ 1 \leq x \leq 2 \quad \text{and} \quad 0 \leq y \leq -x \+ \2 \]
Horizontally, they're given by: \[ 0 \leq y \leq 1 \quad \text{and} \quad 1 \leq x \leq -y \+ \2 \]
These inequalities define the region inside the triangle by highlighting all possible x and y values within the boundaries.
Boundary Equations
Boundary equations constitute lines or curves that define the edges of a geometric region.
In our triangle, these boundaries are defined by three lines:
The vertical and horizontal lines are straightforward, while the diagonal line comes from connecting points \( (1,1) \) and \( (2,0) \).
You can use the point-slope form to find such a line: \[ y \- \1 \= \frac{0 \- 1}{2 \- 1} \ (x \- 1) \rightarrow y \= -x \+ 2 \]
Understanding these boundary equations is essential for mapping out any region's exact shape and limits.
In our triangle, these boundaries are defined by three lines:
- Vertical line: x = 1
- Horizontal line: y = 0
- Diagonal line: y = -x + 2
The vertical and horizontal lines are straightforward, while the diagonal line comes from connecting points \( (1,1) \) and \( (2,0) \).
You can use the point-slope form to find such a line: \[ y \- \1 \= \frac{0 \- 1}{2 \- 1} \ (x \- 1) \rightarrow y \= -x \+ 2 \]
Understanding these boundary equations is essential for mapping out any region's exact shape and limits.
Other exercises in this chapter
Problem 20
Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the region bounded by \(y=\sqrt{x}\) and \(y=x^{2}\).
View solution Problem 21
Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the rectangle with vertices \((-1,1),(2,1)\), \((2,2)\), and
View solution Problem 23
Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the region bounded by \(y=\ln x, y=0\), and \(x=e\).
View solution Problem 25
Evaluate the given double integral for the specified region \(R\).\(\iint_{R} 3 x y^{2} d A\), where \(R\) is the rectangle bounded by the lines \(x=-1, x=2, y=
View solution