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Chapter 11: Problem 30

Find the equation of the surface that results when the curve \(4 x^{2}-3 y^{2}=12\) in the \(x y\) -plane is revolved about the \(x\) -axis.

Short Answer

Expert verified
The equation of the surface is \(4x^2 - 3z^2 = 12\).

Step by step solution

01

Identify the Curve

The given curve is a hyperbola, described by the equation \(4x^2 - 3y^2 = 12\). This is a horizontal hyperbola centered at the origin in the \(xy\)-plane.
02

Solve for y in terms of x

To revolve the curve around the \(x\)-axis, first solve for \(y\). Rearrange the equation \(4x^2 - 3y^2 = 12\) to isolate \(y^2\): \[-3y^2 = 12 - 4x^2\] or \[y^2 = \frac{4x^2 - 12}{-3} = \frac{4x^2 - 12}{3}\].
03

Consider Revolution Around the x-axis

When a curve is revolved around the \(x\)-axis, the resulting surface is defined by \(x\) and the radius \(y\). Here, \(z\) represents values below and above the \(x\)-axis and \(z^2\) will replace \(y^2\): \(z^2 = \frac{4x^2 - 12}{3}\).
04

Formulate the Surface Equation

Replace \(y^2\) with \(z^2\) to get the equation of the surface of revolution: \[4x^2 - 3z^2 = 12\]. This is the standard form of the hyperbolic cylinder obtained upon revolving the hyperbola around the \(x\)-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface of Revolution
A surface of revolution is a three-dimensional surface obtained by rotating a curve around a straight line, known as the axis of revolution. The process is like spinning an object around a string or a rod, where the curve generates a surface as if it left a trail while spinning.
  • The axis of revolution can be the x-axis, y-axis, or any other line in the coordinate plane.
  • A well-known example is when a curve is revolved around an axis. The curve's every point sweeps out a circle with the axis as the center when it's rotated around that axis.
  • This helps to visualize and understand solid figures in terms of simpler two-dimensional shapes.
When you take a plane curve and revolve it, every point on the curve moves in a circle in a plane perpendicular to the axis of revolution. The overall shape of the surface depends on the curve's equation and the chosen axis.
Revolving Curves
Revolving curves is the mathematical procedure where a two-dimensional shape or curve is turned around a line (usually an axis) to create a three-dimensional object.
  • This is a foundational concept in advanced geometry and calculus, enabling the calculation of volumes and surface areas of complex shapes by simplifying them to known forms.
  • The result of revolving curves is often shapes like cylinders, cones, or spheres, depending on the original curve's nature.
  • For a hyperbola, as in our exercise, revolving it about the x-axis produces a hyperbolic cylinder.
The mathematical approach typically involves expressing the curve's equation in terms of the axis and then forming a new equation to represent the surface in three dimensions. This revolves around replacing components in the curve's equation with three-dimensional equivalents to account for the added dimension created by the revolution.
Hyperbola
A hyperbola is a type of conic section or two-dimensional curve from the intersection of a plane with both halves of a double cone. It comprises two disconnected curves called branches that resemble mirrored bows or the stylized wings of a butterfly.
  • Each branch of a hyperbola can be thought of as creating arch-like paths, opening in opposite directions from a central point known as the center.
  • The general equation of a hyperbola is \(Ax^2 - By^2 = C\) or \(Ay^2 - Bx^2 = C\), distinguishing it from other conics like ellipses and parabolas.
  • In the provided exercise, the curve \4x^2 - 3y^2 = 12\ represents a hyperbola in the x-y plane, centered at the origin.
When revolved around an axis like the x-axis, it generates a hyperbolic cylinder, as the two branches extend and create a continuous surface. This functionally becomes a surface of revolution, illustrating the elegant shift from two-dimensional to three-dimensional geometry.

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