Question

In Exercises 31-36, find an equation for the surface of revolution generated by revolving the curve in the indicated coordinate plane about the given axis. \(z=2 y\) \(y z\) -plane \(z\) -axis

Short answer

The equation for the surface of revolution generated by the curve \(z = 2y\) about the \(z\)-axis is \(x^{2} = \frac{z^{2}}{4}\) - \(y^{2}\).

Step-by-step solution

Understanding the concept

First, one needs to get a grasp of the concept of rotation about an axis. In this case, revolving the curve \(z = 2y\) about the \(z\)-axis will generate a paraboloid. The distance to the \(z\)-axis at any point on the plane will be the radius of the rotation, which is \(y\). The new shape will still be symmetric with respect to the \(z\)-axis.

Applying the Pythagorean theorem

The equation for a point on the surface of revolution can be derived using the Pythagorean theorem. For a point that was originally at a distance \(y\) away from the \(z\)-axis, after the rotation, the distance from the \(z\)-axis will remain as \(y\) but spread over two dimensions (\(x, y\)), and \(z\) will remain the same. Therefore the equation of the rotated curve will be \(x^{2}+y^{2}\) in 2D plane, which is derived from \(r^{2}=x^{2}+y^{2}\), where \(r\) is the radius of the rotation, and \(z = 2y\) to express \(z\) as a function of \(y\).

Substituting to find the final equation

Since \(x^{2}+y^{2}=r^{2}=y^{2}\), it becomes \(x^{2}+y^{2} = y^{2}\). Also, given the curve \(z = 2y\), substituting \(y\) into the equation, it becomes \(x^{2} + \frac{z^{2}}{4} = \frac{z^{2}}{4}\). Simplifying, the final equation becomes \(x^{2} = \frac{z^{2}}{4}\) - \(y^{2}\).

Key concepts

Parametric Equations

When we deal with curves in mathematics, particularly those involved in surfaces of revolution, parametric equations become an invaluable tool. They allow us to express curves using parameters – typically variables like t or s – which represent quantities that can systematically take on different values. In essence, parametric equations split a curve into its constituent parts (such as x, y, and z coordinates) and describe how these parts vary together.

For example, a circle in a plane can be described parametrically as x = r cos(t) and y = r sin(t), where r is the radius and t is the parameter that ranges from 0 to 2π radians. This method extends to 3D space for more complex curves, such as the one in our exercise, which upon revolution about an axis, describe a solid of revolution like a paraboloid.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the sides of a right-angled triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is represented as a² + b² = c², where c is the length of the hypotenuse.

When working on surfaces of revolution, this theorem is key in determining the relationships between points before and after the revolution. It helps us to derive equations for the surfaces by considering the radial distances from the axis of revolution to any point on the curve. In building up to the solution in the exercise, we used the Pythagorean theorem to expand the original equation after rotating our curve to encompass the new x-dimension, resulting in x² + y² = r².

Solid of Revolution

A solid of revolution is a three-dimensional object created by rotating a two-dimensional curve around an axis. The resulting shape has a symmetry around the axis of rotation. Visualize taking a piece of paper with a curve drawn on it and spinning it around one of its edges – the shape traced out in space would be a solid of revolution.

The curve in our exercise, z = 2y, when rotated around the z-axis, generates a three-dimensional shape known as a paraboloid. By applying the concept of solids of revolution to this curve, we can derive an equation that represents every single point on the surface of this newly formed shape.

Paraboloid

A paraboloid is a specific type of solid of revolution generated when a parabola is revolved around its axis of symmetry. There are two kinds of paraboloids: elliptic and hyperbolic, depending on the nature of the cross-sectional curve. An elliptic paraboloid, like the one produced in this exercise, is obtained by revolving a parabola around its axis.

The equation of a paraboloid can often be expressed in the general form z = Ax² + By² (where A and B are constants that determine the 'stretch' of the paraboloid), but our specific exercise simplifies to x² = z²/4 - y² after we substituted the revolution's radius and the condition z = 2y. This clean and simple equation effectively describes a three-dimensional, 'bowl-shaped' surface that opens upwards.