Problem 32 - Thomas Calculus

Question

Parametrization of a surface of revolution Suppose that the parametrized curve \(C :(f(u), g(u))\) is revolved about the \(x\) -axis, where \(g(u)>0\) for \(a \leq u \leq b .\) a. Show that $$ \mathbf{r}(u, v)=f(u) \mathbf{i}+(g(u) \cos v) \mathbf{j}+(g(u) \sin v) \mathbf{k} $$ is a parametrization of the resulting surface of revolution, where \(0 \leq v \leq 2 \pi\) is the angle from the \(x y\) -plane to the point \(\mathbf{r}(u, v)\) on the surface. (See the accompanying figure.) Notice that \(f(u)\) measures distance along the axis of revolution and \(g(u)\) measures distance from the axis of revolution. b. Find a parametrization for the surface obtained by revolving the curve \(x=y^{2}, y \geq 0,\) about the \(x\) -axis.

Step-by-Step Solution

Answer

The surface is parametrized by \(\mathbf{r}(u, v) = u^2 \mathbf{i} + (u\cos v) \mathbf{j} + (u\sin v) \mathbf{k}\).

Step 1: Identify Parametric Curve

The given curve is parametrized as \(C: (f(u), g(u))\). This is the base curve that will be rotated about the \(x\)-axis to form the surface of revolution.

Step 2: Parametric Form after Revolution

To parametrize the surface formed by revolving the curve \((f(u), g(u))\) around the \(x\)-axis, use the parameterization formula given by \(\mathbf{r}(u, v) = f(u)\mathbf{i} + (g(u)\cos v)\mathbf{j} + (g(u)\sin v)\mathbf{k}\) where \(v\) is the angle of rotation from the \(xy\)-plane.

Step 3: Understanding Each Component

- \(f(u)\mathbf{i}\) places points along the \(x\)-axis, the axis of revolution.- \(g(u)\cos v\mathbf{j}\) and \(g(u)\sin v\mathbf{k}\) describe a rotation in the \(yz\)-plane, forming a circular cross-section about the \(x\)-axis.

Step 4: Parametrize Specific Curve

The given curve equation is \(x = y^2\), where \(y \geq 0\). To find a suitable parameter \(u\), let \(u = y\), which implies that \(x = u^2\). Thus the curve is parameterized as \((u^2, u)\).

Step 5: Substitute in Parametric Form

Replace \(f(u)\) and \(g(u)\) with the parametric equations found in Step 4: \(f(u) = u^2\) and \(g(u) = u\). The parametrization of the surface becomes \(\mathbf{r}(u, v) = u^2 \mathbf{i} + (u\cos v) \mathbf{j} + (u\sin v) \mathbf{k}\).

Key Concepts

Surface of Revolution

A surface of revolution is formed when a curve is revolved around a fixed line, known as the axis of revolution. This technique is popular in calculus and physics for generating surfaces and solids with rotational symmetry. It can be visualized as folding a 2D curve and spinning it about an axis to create a 3D shape.
For example, consider revolving a semicircle about the x-axis. You would generate a sphere. In our case, the curve described by \(f(u), g(u)\) is revolved around the x-axis, creating a 3D surface. The function \(f(u)\) is responsible for the distance along the axis, while \(g(u)\) determines how far each point on the curve is from the axis. This behavior is crucial because it dictates the shape formed during revolution, impacting both surface area and volume calculations. Understanding surface of revolution helps in visualizing complex shapes formed in engineering and various applied fields.

Parametric Equations

Parametric equations allow for the description of geometric objects by expressing the coordinates as functions of a parameter. This method provides greater flexibility to describe complex shapes and motions that are not easily defined by single equations.

• For a single curve, a parametric form \(\text{like } C: (f(u), g(u))\)\