Question

Sketch the surface in 3 -space. $$z=1-y^{2}$$

   Sketch the surface in 3 -space.
$$z=1-y^{2}$$

Calculus A New Horizon

Calculus A New Horizon

Howard Anton 6th Edition

Chapter 13, Problem 27

Instant Answer

Solved by Expert Linh Vu

09/03/2020

Step 1

First, let's consider the cross-sections of the surface along the x-axis. Since there is no x term in the equation, the cross-sections will be the same for all values of x. So, we can just focus on the yz-plane. Show more…

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Sketch the surface in 3 -space. $$z=1-y^{2}$$

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Key Concepts

Explicit Surface Representation in 3D

This concept pertains to specifying a surface by expressing one coordinate as a function of the other two, which aids in analyzing and visualizing the surface's behavior in three-dimensional space. It sets the framework for understanding how changing variables affect the overall shape of the surface.

Parabolic Cylinder

A parabolic cylinder is a type of surface where the cross-section in one plane is a parabola and the surface extends infinitely in a direction perpendicular to the plane of the parabola. In the equation given, the absence of the x-variable indicates that every cross-section parallel to the x-axis is the same, forming a cylinder whose rulings are parallel to the x-axis.

Cross-Sectional Analysis

This involves examining slices of the surface by fixing one coordinate, which simplifies the visualization of complex three-dimensional structures. For example, by analyzing the y-z plane by setting a constant value for x, one can see the parabolic curves described by z = 1 - y^2, thereby understanding how the surface behaves along different cross-sections.

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