Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

The hyperbolic paraboloid \(z=y^{2}-x^{2}\) over the region (a) in the first quadrant and inside the circle \(x^{2}+y^{2}=9\) (b) inside the triangle with vertices \((0,0),(3,0),(0,3)\)

Step-by-Step Solution

Answer

Solve \(z = y^2 - x^2\) for part (a) within the circle and for part (b) within the triangle's bounds.

Step 1: Understand the Region for Part (a)

For part (a), we are concerned with the region in the first quadrant where both \(x\) and \(y\) are non-negative and are inside the circle defined by \(x^2 + y^2 = 9\). This means we need to solve the equation considering values of \(x\) and \(y\) such that: \(x^2 + y^2 \leq 9\) and \(x \geq 0, y \geq 0\).

Step 2: Solve for Part (a) within Circle Constraints

In polar coordinates, the circle can be described by \(r \leq 3\) where \(r = \sqrt{x^2+y^2}\). The first quadrant means the angles \(0 \leq \theta \leq \frac{\pi}{2}\). Convert \(z = y^2 - x^2\) into polar coordinates using \(x = r\cos\theta\) and \(y = r\sin\theta\). Then solve \(z = (r\sin\theta)^2 - (r\cos\theta)^2 = r^2(\sin^2\theta - \cos^2\theta)\).

Step 3: Understand the Region for Part (b)

For part (b), consider the triangle with vertices at \((0,0)\), \((3,0)\), and \((0,3)\). This triangular region is defined by the inequalities \(0 \leq x \leq 3\), \(0 \leq y \leq 3 - x\).

Step 4: Solve for Part (b) within Triangle Constraints

Within the triangular region, to solve for \(z = y^2 - x^2\), plug in the bounds of the triangular region into the equation. Notice that the function's behavior is controlled by the values along the line \(y = 3 - x\) and between \(x = 0\) and \(y = 0\). Evaluate how \(y^2 - x^2\) behaves at various points that satisfy the triangle's constraints.

Key Concepts

Polar Coordinates

Polar coordinates are an alternative coordinate system to the standard Cartesian coordinates. Instead of using two axes as a grid (like x and y), polar coordinates use the distance from the origin (r) and the angle from the positive x-axis (θ) to define a point in the plane.
This system is beneficial when dealing with circular or radial symmetry.

• The distance, r, is given by the formula: \[ r = \sqrt{x^2 + y^2} \]
• The angle, θ, is defined as the direction measured from the positive x-axis, and it can vary from \(0\) to \(2\pi\) radians.

In the exercise for the hyperbolic paraboloid in part (a), the circle \(x^2 + y^2 = 9\) transforms into \(r \leq 3\) in polar coordinates. We convert points using the equations \(x = r\cos\theta\) and \(y = r\sin\theta\). This simplifies problems involving circles or rotational dynamics.

First Quadrant

The first quadrant is one of the four sections of the Cartesian coordinate plane. It covers the region where both x and y are non-negative, specifically where \(x \geq 0\) and \(y \geq 0\).
Why is this important? Many geometric and mathematical problems often need solutions bounded to this quadrant to ensure only real-world applicable results are considered.

• In the given exercise part (a), the region is not only bounded by a circle defined by \(x^2 + y^2 \leq9\) but must remain in the first quadrant, hence \(0 \leq \theta \leq \frac{\pi}{2}\).

By ensuring the solution is confined to the first quadrant, we only consider the non-negative range of angles and solutions, which often makes interpreting results simpler and more applicable.

Triangle Region

The triangle region in the exercise's part (b) is defined using three points: \((0,0)\), \((3,0)\), and \((0,3)\). This triangle lies within the first quadrant, which is crucial for boundary-driven solutions.

• Its boundaries are easily described by its vertices and the linear inequality \(0 \leq y \leq 3 - x\), where x ranges between \(0\) and \(3\).

Visualizing the triangle helps us understand the limits within which the function \(z = y^2 - x^2\) should be evaluated. Solving problems in triangular areas often involves expanding or manipulating these inequalities to explore corner cases and boundaries.

Boundary Constraints

Boundary constraints are limits on the values that a variable or set of variables can take based on real-world scenarios or specific conditions set by the problem.
Constraints help define the region of interest in a mathematical problem.

• For region (a) of the exercise, the circle's boundary \(x^2 + y^2 = 9\) interacts with x and y's non-negativity, creating a quarter circle.
• For region (b), it is the lines and points of the triangle that bind the scope of the investigation.

Exploring how these boundaries guide the behavior of functions like \(z = y^2 - x^2\) is crucial as they directly influence possible solutions and applicable integrations or evaluations.