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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.

3.C yz cos x ds, C: x = t, y = 3 cos t, z = 3 sin t, 0 ⩽ t ⩽ π

To determine

To Evaluate: The line integral Cyzcosxds C:x=t,y=3cost,z=3sint,0tπ .

Explanation

Given data:

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

C:x=t,y=3cost,z=3sint,0tπ

Formula used:

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

Cf(x,y)ds=abf(x(t),y(t))(dxdt)2+(dydt)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 and integration formulae to evaluate the given integral.

ddxcosx=sinxddxsinx=cosx[f(x)]ndx=[f(x)]n+1n+1

Modify equation (1) as follows.

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

Substitute yzcosx for f(x,y) , t for x , 3cost for y , 3sint for z , 0 for a , and π for b ,

Cyzcosxds=0π(3cost)(3sint)(cost)(ddt(t))2+(ddt(3cost))2+(ddt(3sint))2dt=0π(9cos2tsint)(1)2+(3sint)2+(3cost)2dt=0π(9cos2tsint)1+9sin2t+9cos2tdt=0π(9cos2tsint)1+9(sin2t+cos2t)dt

Rewrite the equation as follows

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Chapter 16 Solutions

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