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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Evaluate the line integral ∫C F · dr, where C is given by the vector function r(t).

20. F(x, y, z) = (x + y2) i + xz j + (y + z) k,

r(t) = t2i + t3j – 2t k, 0 t 2

To determine

To Evaluate: The line integral CFdr .

Explanation

Given data:

The continuous vector field and the vector function are given as follows.

F(x,y,z)=(x+y2)i+xzj+(y+z)kr(t)=t2i+t3j2tk,0t2

Formula used:

Write the expression to evaluate the line integral of vector field F(x,y,z) along the curve C .

CFdr=abF(r(t))r(t)dt (1)

Here,

r(t) is the vector function,

a is the lower limit of curve C , and

b is the upper limit of the curve C .

Write the vector function as follows.

r(t)=t2i+t3j2tk

Write the point (x,y,z) from the vector function as follows.

(x,y,z)=(t2,t3,2t)

Write the vector field as follows.

F(x,y,z)=(x+y2)i+xzj+(y+z)k (2)

Calculation of F(r(t)) :

Substitute t2 for x , t3 for y , (2t) for z in equation (2),

F(r(t))=[t2+(t3)2]i+[t2(2t)]j+[t3+(2t)]k=(t2+t6)i+[(2t3)]j+(t32t)k=(t2+t6)i2t3j+(t32t)k

Calculation of r(t) :

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function r(t)=t2i+t3j2tk as follows.

ddt[r(t)]=ddt(t2i+t3j2tk)

Rewrite the expression as follows

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