Question

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 \(y\) -axis.

Short answer

The surface equation is \(x^2 + z^2 = \frac{12 - 3y^2}{4}\).

Step-by-step solution

Identify the Given Curve

We start with the given equation of the curve: \(4x^2 + 3y^2 = 12\). This is an ellipse centered at the origin in the \(xy\)-plane.

Solve for x in terms of y

To express \(x\) in terms of \(y\), rearrange the equation as follows: \(4x^2 = 12 - 3y^2\), leading to \(x^2 = \frac{12 - 3y^2}{4}\).

Set up the Solid of Revolution Formula

When a curve is revolved around the \(y\)-axis, the resulting surface is given by revolving the function \(x = \sqrt{\frac{12 - 3y^2}{4}}\) about the \(y\)-axis.

Convert to Cylindrical Coordinates

In cylindrical coordinates, \(x = r\cos(\theta)\) and \(r^2 = x^2 + z^2\). The equation \(x^2 = \frac{12 - 3y^2}{4}\) is transformed to \(x^2 + z^2 = \frac{12 - 3y^2}{4}\), where \(z\) covers the rotation.

Write the Surface Equation

The equation of the surface resulting from revolving the curve is \(x^2 + z^2 = \frac{12 - 3y^2}{4}\), where \(y\) represents the vertical axis in this context.

Key concepts

Elliptical Cross-section

In geometry, an ellipse is a shape that resembles a flattened circle. It is defined by an equation where the sum of the distances to two fixed points, called foci, is constant. For this particular problem, the cross-section of the surface of revolution is an ellipse, given by the equation \(4x^2 + 3y^2 = 12\). This equation represents an ellipse centered at the origin in the \(xy\)-plane. An ellipse will appear as an oval shape and is characterized by its major axis (the longest diameter) and its minor axis (the shortest diameter).

The formula above represents a standard form of an ellipse with coefficients defining its shape. The larger the coefficient associated with a variable, the shorter the corresponding axis. In this case:

• The coefficient 4 relates to \(x^2\), representing the squashing of the ellipse along the \(x\)-axis.
• The coefficient 3 relates to \(y^2\), influencing the stretching along the \(y\)-axis.

Understanding elliptical cross-sections allows us to visualize how curves transform when revolved.

Cylindrical Coordinates

Cylindrical coordinates are an extension of the two-dimensional polar coordinate system to three dimensions. They are particularly useful here because they simplify the process of representing the surface resulting from revolving an object's cross-section around an axis.

This system involves three variables:

• \(r\): the radial distance from the origin to the point's projection in the \(xy\)-plane.
• \(\theta\): the angle between the positive \(x\)-axis and the line from the origin to the point's projection.
• \(z\): the height above the \(xy\)-plane.

The original Cartesian equation \(x^2 = \frac{12 - 3y^2}{4}\) converts into cylindrical form because the rotation aspect joins with a vertical axis, creating the term \(x^2 + z^2\) in the equation. This transformation facilitates solving and visualizing the problem.

Solid of Revolution

The concept of a solid of revolution involves creating a three-dimensional shape by rotating a two-dimensional curve around an axis. This axis acts as a pivot point, creating a symmetrical and typically smooth 3D object called a solid of revolution.

For this problem, revolving the ellipse defined by \(4x^2 + 3y^2 = 12\) around the \(y\)-axis forms this solid.

• Visualize this by imagining the ellipse spinning 360 degrees around the vertical \(y\)-axis, forming a shape similar to a discus or a vase.
• The constant distribution of points due to rotation results in the surface equation \(x^2 + z^2 = \frac{12 - 3y^2}{4}\), representing both the shaping and dimensions of this surface.

This technique helps visualize complex equations by transforming them into tangible forms.

Coordinate Transformation

Coordinate transformation refers to the process of changing one coordinate system to another. Here, it involves transforming Cartesian coordinates into cylindrical coordinates. Such transformation makes the problem easier to manage, especially in three-dimensional space.

The original ellipse centered on the \(xy\)-plane in Cartesian form, \(4x^2 + 3y^2 = 12\), undergoes transformation to a 3D form via coordinate changes. We find:

• Transformation aligns with the change to cylindrical coordinates, where \(x = r\cos(\theta)\), and the equation modifies to account for rotation by including \(z\).
• This conversion is important because it efficiently describes the rotational aspects of the solid created from revolution and represents the symmetrical properties easier to grasp and manipulate.

Understanding this process enhances the ability to interpret complex multi-dimensional shapes in simpler terms.