Problem 20
Question
Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the region bounded by \(y=\sqrt{x}\) and \(y=x^{2}\).
Step-by-Step Solution
Verified Answer
For 0≤x≤1: x²≤y≤√x and for 0≤ y ≤ 1: y²≤x≤√y.
1Step 1 - Identify the given curves
Recognize the curves that bound the region R: the first equation is given by \( y = \sqrt{x} \) and the second equation is \( y = x^2 \).
2Step 2 - Find the points of intersection
Set \( y = \sqrt{x} \) equal to \( y = x^2 \) to find the points of intersection. Solve the equation: \[ \sqrt{x} = x^2 \] Square both sides: \[ x = x^4 \] Rearrange and solve for x: \[ x^4 - x = 0 \] Factor: \[ x(x^3 - 1) = 0 \] Solve for x: \[ x = 0 \] or \( x^3 = 1 \) which gives \[ x = 1 \]
3Step 3 - Determine the range of x-values
The points of intersection occur at \( x = 0 \) and \( x = 1 \). Therefore, the range of x-values for region R is \( 0 \leq x \leq 1 \).
4Step 4 - Describe vertical cross-sections
For any value of x between 0 and 1, the value of y is bounded below by \( y = x^2 \) and bounded above by \( y = \sqrt{x} \). Thus, in terms of y, the inequality describing the vertical cross-section is: \[ x^2 \leq y \leq \sqrt{x} \]
5Step 5 - Describe horizontal cross-sections
To describe in terms of horizontal cross-sections, solve each equation for x in terms of y: For \( y = \sqrt{x} \), solve for x: \[ x = y^2 \] For \( y = x^2 \), solve for x: \[ x = \sqrt{y} \] (considering only the positive square root since x is non-negative). Thus, when 0 ≤ y ≤ 1, we have: \[ y^2 \leq x \leq \sqrt{y} \]
Key Concepts
inequalities in coordinate geometry
inequalities in coordinate geometry
To understand inequalities in coordinate geometry, it's essential to grasp that they help us describe regions in the plane where certain conditions are met. For instance, if we have two curves, we can use inequalities to specify areas where one curve lies above or below the other.
In the exercise, we're working with the curves given by the equations \( y = \sqrt{x} \) and \( y = x^2 \). These curves divide the plane into different regions. By analyzing these inequalities, we can determine which parts of the plane each curve influences.
By understanding the \
In the exercise, we're working with the curves given by the equations \( y = \sqrt{x} \) and \( y = x^2 \). These curves divide the plane into different regions. By analyzing these inequalities, we can determine which parts of the plane each curve influences.
By understanding the \
Other exercises in this chapter
Problem 18
Evaluate the double integrals.\(\int_{0}^{3} \int_{y^{2} / 4}^{\sqrt{10-y^{2}}} x y d x d y\)
View solution Problem 19
Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the region bounded by \(y=x^{2}\) and \(y=3 x\).
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 22
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 \((2,0)\).
View solution