   Chapter 16, Problem 7RE

Chapter
Section
Textbook Problem

Evaluate the line integral.7. ∫C xy dx + y2 dy + yz dz, C is line segment from (1,0, −1), to (3,4, 2)

To determine

To Evaluate: The line integral Cxydx+y2dy+yzdz , C is the line segment from (1,0,1) , to (3,4,2) .

Explanation

Given data:

The line segment is given as follows.

(1,0,1) , to (3,4,2)

Formula used:

Write the expression vector representation of line segment.

r(t)=(1t)r0+tr1 0t1 (1)

Here,

r0 is starting point of line segment, and

r1 is ending point of line segment.

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

CP(x,y)dx+Q(x,y)dy=CP(x,y)dx+CQ(x,y)dy

CP(x,y)dx+Q(x,y)dy=[abP(x(t),y(t))x(t)dt+abQ(x(t),y(t))y(t)dt] (2)

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.

ddxxn=nxn1[f(x)]ndx=[f(x)]n+1n+1

Find parametric representation of line segment for x-coordinate using equation (1).

Substitute x for r(t) , 1 for r0 and 3 for r1 in equation (1),

x=(1t)(1)+t(3)=1t+3t=1+2t

Differentiate the equation with respect to t .

dxdt=0+2dx=2dt

Find parametric representation of line segment for y-coordinate using equation (1).

Substitute y for r(t) , 0 for r0 and 4 for r1 in equation (1),

y=(1t)(0)+t(4)=0+4t=4t

Differentiate the equation with respect to t .

dydt=4dy=4dt

Find parametric representation of line segment for z-coordinate using equation (1).

Substitute z for r(t) , 1 for r0 and 2 for r1 in equation (1),

z=(1t)(1)+t(2)=1+t+2t=1+3t

Differentiate the equation with respect to t

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