Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**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.
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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.
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