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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.

11. C xeyz ds, C is the line segment from (0, 0, 0) to (1, 2, 3)

To determine

To Evaluate: The line integral C(xeyz)ds for the line segment from the point (0,0,0) to the point (1,2,3) .

Explanation

Given data:

The given curve C is a line segment from the point (0,0,0) to the point (1,2,3) .

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 expression to find the parametric equations for a line through the point (x0,y0,z0) and parallel to the direction vector v=a,b,c .

x=x0+at,y=y0+bt,z=z0+ct (2)

Write the expression to find direction vector v=a,b,c for a line segment from the point (x0,y0,z0) to the point (x1,y1,z1) .

a,b,c=x1x0,y1y0,z1z0 (3)

Write the required differential formulae to evaluate the given integral.

ddttn=ntn1

Consider the points (x0,y0,z0) as (0,0,0) and (x1,y1,z1) as (1,2,3) .

Calculation of direction vector:

Substitute 1 for x1 , 2 for y1 , 3 for z1 , 0 for x0 , 0 for y0 , and 0 for z0 in equation (3),

a,b,c=10,20,30=1,2,3

Calculation of parametric equations of the curve:

Substitute 0 for x0 , 0 for y0 , 0 for z0 , 1 for a , 2 for b , 3 for c in equation (2),

x=0+(1)t,y=0+(2)t,z=0+(3)tx=t,y=2t,z=3t

Consider the limits of scalar parameter t are 0 to 1.

0t1

Find the expression (xeyz) as follows

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Chapter 16 Solutions

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Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 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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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