Question

Sketch the surface in 3 -space. $$y=\sin x$$

   Sketch the surface in 3 -space.
$$y=\sin x$$

Calculus A New Horizon

Calculus A New Horizon

Howard Anton 6th Edition

Chapter 13, Problem 25

Instant Answer

Solved by Expert Linh Vu

09/03/2020

Step 1

First, we need to understand what the equation y = sin(x) represents. This is a 2D curve in the xy-plane, where the y-coordinate of each point on the curve is equal to the sine of the x-coordinate. Show more…

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Sketch the surface in 3 -space. $$y=\sin x$$

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Key Concepts

Three-Dimensional Coordinate System

This concept involves understanding how points and curves are represented in 3D space using three axes—typically labeled as x, y, and z. It is key in visualizing and sketching surfaces because it allows one to see how a curve or surface extends and behaves in the three-dimensional environment.

Extrusion of a Curve

When a curve defined in a two-dimensional plane (such as y = sin x) is extended in a direction independent of the defining variables (in this case, z), it creates a surface. This process, known as extrusion, results in a surface that is essentially a repetition of the curve along the extruded axis.

Cylindrical Surfaces

A cylindrical surface in 3D space is formed when the equation of a surface does not depend on one of the coordinates. In the case where y = sin x, since the expression does not involve z, the graph of the sine function is repeated along the z-axis, resulting in a cylindrical surface.

Sine Function

The sine function is a periodic and oscillatory function that produces wave-like patterns. When used in the equation y = sin x, it provides a repetitive up-and-down fluctuation of y-values as x varies, which forms the distinctive ridge and trough pattern in the surface when extended along the z-axis.

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