Chapter 16.2, Problem 12E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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

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