   Chapter 16.2, Problem 12E

Chapter
Section
Textbook Problem

Evaluate the line integral, where C is the given curve.12. ∫C(x2 + y2 + z2) ds, C: x = t, y = cos 2t, z = sin 2t, 0 ⩽ t ⩽ 2π

To determine

To Evaluate: The line integral C(x2+y2+z2)ds for a curve.

Explanation

Given data:

The parametric equations of curve and its limits are given as follows.

C:x=t,y=cos2t,z=sin2t,0t2π

Formula used:

Write the expression to evaluate the line integral for a function f(x,y,z) along the curve C .

Cf(x,y,z)ds=abf(x(t),y(t),z(t))(dxdt)2+(dydt)2+(dzdt)2dt (1)

Here,

a is the lower limit of the curve C and

b is the upper limit of the curve C .

Write the required differential formulae to evaluate the given integral.

ddttn=ntn1ddtcosnt=nsinntddtsinnt=ncosnt

Calculation of expression (x2+y2+z2) :

Substitute t for x , cos2t for y , sin2t for z in the expression (x2+y2+z2) ,

x2+y2+z2=t2+(cos2t)2+(sin2t)2=t2+cos22t+sin22t=t2+1 {cos2θ+sin2θ=1}=1+t2

Evaluation of line integral C(x2+y2+z2)ds :

Substitute (x2+y2+z2) for f(x,y,z) , (1+t2) for f(x(t),y(t),z(t)) , t for x , cos2t for y , sin2t for z , 0 for a , and 2π for b in equation (1),

C(x2+y

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 the coordinates of the points shown.

Precalculus: Mathematics for Calculus (Standalone Book)

Evaluate the integral. 042sds

Single Variable Calculus: Early Transcendentals, Volume I

Evaluate the integral. 0/2cos2d

Calculus (MindTap Course List)

For y = 3x + tan x, y = _____. a) 3 + sec x b) 3x sec2 x + 3 tan x c) 3 + sec2 x d) x cos x + sin x

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