   Chapter 16, Problem 9RE

Chapter
Section
Textbook Problem

Evaluate the line integral.9. ∫C F · dr, where F(x,y,z) = ez i + xz j + (x + y) k and C is given by r(t) = t2 i + t3 j – t k, 0 ⩽ t ⩽ 1

To determine

To Evaluate: The line integral CFdr , where F(x,y,z)=ezi+xzj+(x+y)k and C is given by r(t)=t2i+t3jtk , 0t1 .

Explanation

Given data:

F(x,y,z)=ezi+xzj+(x+y)k , r(t)=t2i+t3jtk , 0t1 .

Formula used:

Write the expression for r(t) in terms of x , y and z .

r(t)=xi+yj+zk (1)

Write the required differential formula to evaluate the given integral.

ddxxn=nxn1

Write the required integration formula to evaluate the given integral.

[f(x)]ndx=[f(x)]n+1n+1

Consider the expression for r(t) as follows.

r(t)=t2i+t3jtk (2)

Compare equations (1) and (2).

x=t2y=t3z=t

Consider the expression for F(x,y,z) as follows.

F(x,y,z)=ezi+xzj+(x+y)k

Substitute t2 for x , t3 for y and t for z ,

F(x,y,z)=eti+[(t2)(t)]j+(t2+t3)k=etit3j+(t2+t3)k

Differentiate equation (2) with respect to t .

dr=(2ti+3t2jk)dt

Find the value of CFdr .

CFdr=C(etit3j+(t2+t3)k)(2ti+3t2jk)dt

Change the limits with respect to t

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