/*! 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 73 Surface Area of a Torus Find the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 73

Surface Area of a Torus Find the surface area of the torus generated by revolving the circle given by \(r=2\) about the line \(r=5 \sec \theta .\)

Short Answer

Expert verified
The surface area of the torus is \(A = 2\pi^2 [39+32 \sqrt{5}]\).

Step by step solution

01

Conversion to Polar Coordinates

First, we express the circle in terms of polar coordinates, \(r\) and \(\theta\), where \(r=\sqrt{x^2+y^2}\) and \(\theta= \tan^{-1}(y/x)\). This circle is represented by \(f(r,\theta) = r\). Given the initial radius r=2 and the given condition of revolution: \(r = 5 \sec(\theta)\), the area A of the surface of revolution (torus) can be defined in polar coordinates with the following integral, where \(R\) is the total radius of revolution, \(R = r + f(r,\theta)\):
02

Computing the Integrals

Substituting the given values into the integral, we get the equation \(A = \int_0^{2\pi} \int_0^{2\pi}(2+5 \sec \theta) \sqrt{1+(2 \cdot \tan \theta)^2} d\theta dr\). By switching the order of integration, this equation can be broken down into two separate integrals which makes it much easier to solve.
03

Solving the Integrals

Solving the inner integral, we get \(\int_0^{2\pi}(2+5 \sec \theta) \sqrt{1+(2 \cdot \tan \theta)^2} d\theta = \pi [39+32 \sqrt{5}]\). Now, by introducing this back into the outer integral, \(\int_0^{2\pi} \pi[39+32 \sqrt{5}] dr\), we can now integrate with respect to \(dr\).
04

Final Simplification

Finally, simplifying \(\int_0^{2\pi} \pi[39+32 \sqrt{5}] dr\), gives us the final solution for the surface area of the torus: \(A = 2\pi^2 [39+32 \sqrt{5}]\).

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!

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

Area of a Region In Exercises \(57-60\) , use the integration capabilities of a graphing utility to approximate, to two decimal places, the area of the region bounded by the graph of the polar equation. $$ r=\frac{9}{4+\cos \theta} $$

Identifying Conics Identify each conic. $$ \begin{array}{ll}{\text { (a) } r=\frac{5}{1-2 \cos \theta}} & {\text { (b) } r=\frac{5}{10-\sin \theta}} \\ {\text { (c) } r=\frac{5}{3-3 \cos \theta}} & {\text { (d) } r=\frac{5}{1-3 \sin (\theta-\pi / 4)}}\end{array} $$

Finding Points of Intersection In Exercises \(25-32,\) find the points of intersection of the graphs of the equations. $$ \begin{array}{l}{r=2-3 \cos \theta} \\ {r=\cos \theta}\end{array} $$

Sketching and Identifying a Conic In Exercises \(13-22\) , find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $$ r=\frac{5}{-1+2 \cos \theta} $$

Finding the Arc Length of a Polar Curve In Exercises \(51-56,\) find the length of the curve over the given interval. $$ \begin{array}{ll}{\text { Polar Equation }} & {\text { Interval }} \\\ {r=1+\sin \theta} & {0 \leq \theta \leq 2 \pi}\end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Statistics

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.