ind the volume of the solid that lies under the hyperbolic paraboloid z = 3y2 - x2 + 4 and above the rectangle R = [-1, 1] × [1, 3].

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
icon
Related questions
Question
**Problem Statement:**

Find the volume of the solid that lies under the hyperbolic paraboloid defined by the equation \( z = 3y^2 - x^2 + 4 \) and above the rectangle \( R = [-1, 1] \times [1, 3] \).

**Explanation:**

In this problem, we are tasked with calculating the volume of a solid that is bounded above by a hyperbolic paraboloid surface and below by a rectangular region in the xy-plane. 

- **Surface Equation:** The surface, a hyperbolic paraboloid, is given by \( z = 3y^2 - x^2 + 4 \). This is a saddle-shaped surface which can open upwards in one direction and downwards in another.

- **Rectangular Region:** The rectangle \( R \) in the xy-plane is defined by the product of intervals \([-1, 1]\) for x-axis and \([1, 3]\) for y-axis.

**Approach:**

To find the volume of the solid, one typically sets up a double integral over the region \( R \) with the function \( z = 3y^2 - x^2 + 4 \) being integrated. The integration would be:

\[
V = \int_{1}^{3} \int_{-1}^{1} (3y^2 - x^2 + 4) \, dx \, dy
\]

**Diagram Explanation:**

- **Graph of the Hyperbolic Paraboloid:** Though not shown here visually, imagine a 3D graph where the surface fluctuates with the saddle-like curvature that extends over the given rectangular region in the plane.
- **Rectangular Domain in Plane:** The rectangle serves as the projection of the 3D solid onto the 2D plane which determines the limits of integration.

By solving this integral, one determines the volume of the solid bound by these geometric constraints.
Transcribed Image Text:**Problem Statement:** Find the volume of the solid that lies under the hyperbolic paraboloid defined by the equation \( z = 3y^2 - x^2 + 4 \) and above the rectangle \( R = [-1, 1] \times [1, 3] \). **Explanation:** In this problem, we are tasked with calculating the volume of a solid that is bounded above by a hyperbolic paraboloid surface and below by a rectangular region in the xy-plane. - **Surface Equation:** The surface, a hyperbolic paraboloid, is given by \( z = 3y^2 - x^2 + 4 \). This is a saddle-shaped surface which can open upwards in one direction and downwards in another. - **Rectangular Region:** The rectangle \( R \) in the xy-plane is defined by the product of intervals \([-1, 1]\) for x-axis and \([1, 3]\) for y-axis. **Approach:** To find the volume of the solid, one typically sets up a double integral over the region \( R \) with the function \( z = 3y^2 - x^2 + 4 \) being integrated. The integration would be: \[ V = \int_{1}^{3} \int_{-1}^{1} (3y^2 - x^2 + 4) \, dx \, dy \] **Diagram Explanation:** - **Graph of the Hyperbolic Paraboloid:** Though not shown here visually, imagine a 3D graph where the surface fluctuates with the saddle-like curvature that extends over the given rectangular region in the plane. - **Rectangular Domain in Plane:** The rectangle serves as the projection of the 3D solid onto the 2D plane which determines the limits of integration. By solving this integral, one determines the volume of the solid bound by these geometric constraints.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning