/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 Parametrization of a surface of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 32

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.

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.
04

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 \((u^2, u)\).
05

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

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 \(\text{like } C: (f(u), g(u))\)\

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Apply Green's Theorem to evaluate the integrals. \(\oint_{C}\left(y^{2} d x+x^{2} d y\right)\) \(C :\) The triangle bounded by \(x=0, x+y=1, y=0\)

Gradient of a line integral Suppose that \(\mathbf{F}=\nabla f\) is a conservative vector field and $$g(x, y, z)=\int_{(0,0,0)}^{(x, y z)} \mathbf{F} \cdot d \mathbf{r}$$ Show that \(\nabla g=\mathbf{F}\).

Gravitational field \begin{equation}\begin{array}{c}{\text { a. Find a potential function for the gravitational field }} \\ {\mathbf{F}=-G m M \frac{x \mathbf{i}+y \mathbf{j}+z \mathbf{k}}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}} \\ {(G, m, \text { and } M \text { are constants })}\\\\{\text { b. Let } P_{1} \text { and } P_{2} \text { be points at distance } s_{1} \text { and } s_{2} \text { from the origin. }} \\ {\text { Show that the work done by the gravitational field in part (a) }} \\ {\text { in moving a particle from } P_{1} \text { to } P_{2} \text { is }}\end{array}\end{equation} \begin{equation}\operatorname{GmM}\left(\frac{1}{s_{2}}-\frac{1}{s_{1}}\right).\end{equation}

Calculate the net outward flux of the vector field $$\mathbf{F}=x y \mathbf{i}+\left(\sin x z+y^{2}\right) \mathbf{j}+\left(e^{x y^{2}}+x\right) \mathbf{k}$$ over the surface \(S\) surrounding the region \(D\) bounded by the planes \(y=0, z=0, z=2-y\) and the parabolic cylinder \(z=1-x^{2}\) .

Integral along different paths Evaluate the line integral \(\int_{C} 2 x \cos y d x-x^{2} \sin y d y\) along the following paths \(C\) in the \(x y\) -plane. \begin{equation}\begin{array}{l}{\text { a. The parabola } y=(x-1)^{2} \text { from }(1,0) \text { to }(0,1)} \\ {\text { b. The line segment from }(-1, \pi) \text { to }(1,0)} \\ {\text { c. The } x \text { -axis from }(-1,0) \text { to }(1,0)} \\ {\text { d. The astroid } \mathbf{r}(t)=\left(\cos ^{3} t\right) \mathbf{i}+\left(\sin ^{3} t\right) \mathbf{j}, 0 \leq t \leq 2 \pi, \text { counterclockwise }} \\ {\text { from }(1,0) \text { back to }(1,0)}\end{array}\end{equation}
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.