Chapter 16.2, Problem 25E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Use a calculator to evaluate the line integral correct to four decimal places.25. ∫C xy arctan z ds, where C has parametric equations x = t2, y = t3, z = t , 1 ⩽ t ⩽ 2

To determine

To Evaluate: The line integral Cxyarctanzds for a curve C:x=t2,y=t3,z=t,1t2 .

Explanation

Given data:

The parametric equations of curve and its limits are given as follows.

C:x=t2,y=t3,z=t,1t2

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 required differential formulae to evaluate the given integral.

ddttn=ntn1ddtt=12t

Calculation of expression (xyarctanz) :

Substitute t2 for x , t3 for y , and t for z in the expression (xyarctanz) ,

xyarctanz=(t2)(t3)arctan(t)=t5tan1(t)

Evaluation of line integral Cxyarctanzds :

Substitute (xyarctanz) for f(x,y,z) , [

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