Chapter 16, Problem 6RE

### Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

Chapter
Section

### Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

# Evaluate the line integral.6. ∫C x y dx + ey dy + xz dz, C is given by r(t) = t4 i + t2 j + t3 k, 0 ⩽ t ⩽ 1

To determine

To Evaluate: The line integral Cxydx+eydy+xzdz , C is given by r(t)=t4i+t2j+t3k,0t1 .

Explanation

Given data:

The equation of r(t) and its limits are given as follows.

r(t)=t4i+t2j+t3k,0t1 (1)

Formula used:

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)

Write the required differential and integration formulae to evaluate the given integral.

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

Write the expression for vector field.

r(t)=xi+yj+zk (3)

Compare equation (1) and (3).

x=t4

Differentiate the equation with respect to t .

dxdt=4t3dx=4t3dt

Compare equation (1) and (3).

y=t2

Differentiate the equation with respect to t .

dydt=2tdy=2tdt

Compare equation (1) and (3).

z=t3

Differentiate the equation with respect to t .

dzdt=3t2dz=3t2dt

Modify equation (2) as follows

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