Question

Give the equation of the surface of revoIution described. Revolve \(y=x^{2}\) about the \(x\) -axis.

Short answer

The equation of the surface is \(z^2 + x^4 = y^2\).

Step-by-step solution

Understand the Surface of Revolution

The surface of revolution is obtained by rotating a curve about an axis. In this exercise, we are rotating the curve defined by the equation \(y = x^2\) around the \(x\)-axis.

Set Up the Equation for the Surface of Revolution

When a curve \(y=f(x)\) is rotated around the \(x\)-axis, the equation becomes \(z^2 + y^2 = r^2\), where \(z\) is the perpendicular distance from the axis and \(r = y\). However, here we should find an expression using all coordinates (x, y, z).

Utilize the Formula for 3D Representation

By rotating \(y = x^2\) around the \(x\)-axis, any point \((x, y)\) on the curve sweeps out a circle. Therefore, any point on the surface can be described in terms of \(x, y, z\) as \(z^2 + (x^2)^2 = y^2\) which simplifies to \(z^2 + x^4 = y^2\).

Combine the Variables into a Single Equation

The equation \(z^2 + x^4 = y^2\) represents all points \((x, y, z)\) on the surface when the curve \(y = x^2\) is revolved around the \(x\)-axis. This is the desired equation for the surface of revolution.

Key concepts

Rotation of Curves

When dealing with the concept of rotating curves, we are transforming a 2D shape into a 3D object. Imagine the curve as a bendy line on a piece of paper. When we rotate this line around an axis (like the spine of a notebook), we create a 3D shape. This is what we call the 'surface of revolution'. In our example, the curve is defined by the equation \(y = x^2\). This parabola, when spun around the \(x\)-axis, forms a 3D shape resembling a bowl or cup.The rotation turns the curve into a continuous set of circles. Each point on the line becomes a circle perpendicular to the axis of rotation. If you picture slicing the shape, you would see circular cross-sections.Understanding this initial rotation step is crucial. It's the basis for getting from a flat line to a full, round object. By grasping this, the next steps in setting up the mathematical representation become clearer.

3D Equations

Once the curve \(y = x^2\) is rotated about the \(x\)-axis, we need to express this new 3D shape using a mathematical equation. The transition from a 2D curve to a 3D object requires considering all three dimensions: \(x\), \(y\), and \(z\).To achieve this, imagine each point \((x, y)\) on the original curve tracing out a circle around the \(x\)-axis. In three dimensions, these points can be expressed as \((x, y, z)\). The distance from the \(x\)-axis (axis of rotation) to the curve in terms of \(z\) and \(y\) is described by the relationship: \[ z^2 + x^4 = y^2 \]This captures all possible locations of points on the surface of the resulting 3D shape.In breaking it down:- \(z\) measures how far the point is from the \(x\)-y plane vertically.- \(x^4\) is a straightforward transformation from \(y = x^2\), now squared, to meet the symmetry of rotation.Expressing the surfaces this way helps in visualizing the full geometry of objects resulting from such operations.

Calculus Concepts

While the conversion of curves into surfaces of revolution uses geometry, calculus provides the tools to handle the process comprehensively. The fundamental calculus concepts involved include integration and transformation.**Integration in Rotations:** When curves are revolved, integration can help derive key properties like volume or surface area of the created shape. The method involves considering infinitesimally small circular slices (think of each slice as a disk) that make up the surface.For rotations around the \(x\)-axis, the integral helps sum up these slivers to provide total volume or surface area. This is seen in formulas like: \[ V = \pi \int_a^b [f(x)]^2 \, dx \] To explore the properties beyond just visual representation, grasp basics of integration in this context.**Transformations and Coordinate System:** Moving a curve into 3D requires understanding how transformations work. We take familiar 2D functions and plot points in a 3D space through rotation, employing the techniques of changing from a simple function in \(y\) to a multi-variable expression including \(z\).Learning to put calculus together with rotations gives you the power to tackle various forms of physical and theoretical problems. It unlocks the deeper understanding needed for engineering, physics, and other scientific applications.