   Chapter 16.6, Problem 56E

Chapter
Section
Textbook Problem

Find the area of the surface with vector equation r(u, v) = ⟨cos3u cos3v, sin3u cos3v, sin3v⟩, 0 ⩽ u ⩽ π, 0 ⩽ v ⩽ 2π. State your answer correct to four decimal places.

To determine

To find: The area of surface with the vector equation r(u,v)=cos3ucos3v,sin3ucos3v,sin3v,0uπ,0v2π .

Explanation

Given data:

The vector equation of the surface is given as follows.

r(u,v)=cos3ucos3v,sin3ucos3v,sin3v,0uπ,0v2π

Formula used:

Write the expression to find the surface area of the plane with the vector equation r(u,v) .

A(S)=D|ru×rv|dA (1)

Here,

ru is the derivative of vector equation r(u,v) with respect to the parameter u and

rv is the derivative of vector equation r(u,v) with respect to the parameter v .

Write the expression to find ru .

ru=u[r(u,v)] (2)

Write the expression to find rv .

rv=v[r(u,v)] (3)

Calculation of ru :

Substitute cos3ucos3v,sin3ucos3v,sin3v for r(u,v) in equation (2),

ru=ucos3ucos3v,sin3ucos3v,sin3v=u(cos3ucos3v),u(sin3ucos3v),u(sin3v)=cos3vu(cos3u),cos3vu(sin3u),sin3vu(1)=cos3v(3cos2usinu),cos3v(3sin2ucosu),sin3v(0)

Simplify the equation.

ru=3cos2usinucos3v,3sin2ucosucos3v,0

Calculation of rv :

Substitute cos3ucos3v,sin3ucos3v,sin3v for r(u,v) in equation (3),

rv=vcos3ucos3v,sin3ucos3v,sin3v=v(cos3ucos3v),v(sin3ucos3v),v(sin3v)=cos3uv(cos3v),sin3uv(cos3v),3sin2vcosv=cos3u(3cos2vsinv),sin3u(3cos2vsinv),3sin2vcosv

Simplify the equation.

rv=3cos3ucos2vsinv,3sin3ucos2vsinv,3sin2vcosv

Calculation of ru×rv :

Substitute 3cos2usinucos3v,3sin2ucosucos3v,0 for ru and 3cos3ucos2vsinv,3sin3ucos2vsinv,3sin2vcosv for rv in the expression ru×rv ,

ru×rv=3cos2usinucos3v,3sin2ucosucos3v,0×3cos3ucos2vsinv,3sin3ucos2vsinv,3sin2vcosv

Rewrite and compute the expression as follows.

ru×rv=|ijk3cos2usinucos3v3sin2ucosucos3v03cos3ucos2vsinv3sin3ucos2vsinv3sin2vcosv|=(|3sin2ucosucos3v03sin3ucos2vsinv3sin2vcosv|i|3cos2usinucos3v03cos3ucos2vsinv3sin2vcosv|j+|3cos2usinucos3v3sin2ucosucos3v3cos3ucos2vsinv3sin3ucos2vsinv|k)

Simplify the expression as follows

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find all possible real solutions of each equation in Exercises 3144. x3x25x+5=0

Finite Mathematics and Applied Calculus (MindTap Course List)

Let f(x) = logb (3x2 2). For what value of b is f(1) = 3?

Single Variable Calculus: Early Transcendentals, Volume I

In Exercises 29-34, rationalize the denominator of each expression. 32. a1a

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Perform the indicated operation for the following. .5031.2575

Contemporary Mathematics for Business & Consumers

An integral for the solid obtained by rotating the region at the right about the x-axis is:

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Prove the idempotent law for conjunction, ppp.

Finite Mathematics for the Managerial, Life, and Social Sciences 