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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

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

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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Chapter 16 Solutions

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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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