Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

If the curve \(z=x^{2}\) in the \(x z\) -plane is revolved about the \(z\) -axis, the resulting surface has equation \(z=x^{2}+y^{2}\), obtained as a result of replacing \(x\) by \(\sqrt{x^{2}+y^{2}}\). If \(y=2 x^{2}\) in the \(x y\) -plane is revolved about the \(y\) -axis, what is the equation of the resulting surface?

Step-by-Step Solution

Answer

The equation is \(y = 2(x^2 + z^2)\).

Step 1: Identifying Initial Equation

The curve given is \(y = 2x^2\). This is a parabola in the \(xy\)-plane.

Step 2: Revolution about the y-axis

The task requires revolving the curve \(y = 2x^2\) around the \(y\)-axis. The distance from a point on the curve to the \(y\)-axis is \(x\).

Step 3: Replacing x with radial distance

To describe the surface of revolution, we replace \(x\) with the radial distance \(r = \sqrt{x^2 + z^2}\), as points vary around the \(y\)-axis.

Step 4: Resulting Surface Equation

Substitute \(x = \sqrt{x^2 + z^2}\) in the original equation \(y = 2x^2\). It becomes \(y = 2(x^2 + z^2)\) leading to \(y = 2r^2\).

Key Concepts

Parabola

A parabola is a symmetrical curve formed by all points that are the same distance from a particular point, known as the focus, and a line called the directrix. In mathematical terms, this can be expressed by quadratic equations such as \(y = ax^2 + bx + c\). In the exercise, the parabola enters the scene with the equation \(y = 2x^2\). This means that for every \(x\) value, the \(y\) is obtained by doubling the square of \(x\).
This particular equation shows us that the parabola opens upwards along the \(y\)-axis, centered at the origin. Parabolas have unique properties:

• Vertex: the turning point, which is the minimum or maximum on the curve.
• Axis of Symmetry: a line that vertically splits the parabola into identical halves.
• Direction: determined by the sign of the coefficient of \(x^2\); in this case, upwards since the coefficient \(2\) is positive.

The understanding of this setup becomes crucial when we start revolving the shape about an axis.

Revolution About an Axis

Revolution about an axis involves rotating a 2D shape or curve around a given axis to create a 3D object. In this scenario, the curve \(y = 2x^2\) is revolved around the \(y\)-axis. Imagine swinging a thin parabola like a baton around its length; the result is a 3D solid known as a surface of revolution.
This transformation process alters how we view and understand the geometry. Some key aspects of revolution include:

• Generating new dimensions: turning a 2D shape (parabola) into a 3D object.
• Axis of rotation: thoroughly influencing the geometry of the resulting shape, dictating symmetry and dimensions.
• Visualizing revolving: akin to imagining the paths traced out by an ever-expanding rubber band around the central axis.

For the problem at hand, a paraboloid results from rotating the initial curve around the \(y\)-axis.

Equation of Surface

To find the equation of the surface resulting from the revolution, we need to incorporate the radial distance involved when revolving around an axis. The given parabola \(y = 2x^2\) becomes the starting point. When this curve revolves around the \(y\)-axis, our main task is to incorporate the \(r\), the radial distance.
Replacing \(x\) in the equation with \(r = \sqrt{x^2 + z^2}\) transforms the equation, accounting for circular symmetry.
The original equation, \(y = 2x^2\), transitions into \(y = 2(x^2 + z^2)\), which is simplified to \(y = 2r^2\). Here, \(r\) acts as a true radial distance from the axis, now symmetrical in 3D form.

Cylindrical Coordinates

Cylindrical coordinates offer a way to simplify the description of systems involving rotational symmetry. The coordinates \((r, \theta, z)\) provide a means to describe every point in 3D just like standard \(x, y, z\) coordinates, but in a cylinder-like approach.

• \(r\) is the radial distance from the origin, or how far out from the axis of rotation (here the \(y\)-axis) a point is.
• \(\theta\) is the angular displacement around the axis, romantically comparable to sweeping around like the hands of a clock.
• \(z\) remains the same, providing vertical measurement.

This adaptation allows us to comprehensively picture the parabolic revolution in a newer coordinate system where symmetry and distance from rotational axis are highlighted. Transformations like \(x = \sqrt{x^2 + z^2}\) emerge naturally when shifting to these cylindrical perspectives, seamlessly linking geometry to algebraic equations as explored previously.