   Chapter 16.2, Problem 26E

Chapter
Section
Textbook Problem

Use a calculator to evaluate the line integral correct to four decimal places.26. ∫C z ln(x + y) ds, where C has parametric equations x = 1 + 3t, y = 2 + t2, z = t4, −1 ⩽ t ⩽ 1

To determine

To Evaluate: The line integral Czln(x+y)ds for a curve C:x=1+3t,y=2+t2,z=t4,1t1 .

Explanation

Given data:

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

C:x=1+3t,y=2+t2,z=t4,1t1

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 formula to evaluate the given integral.

ddttn=ntn1

Calculation of expression [zln(x+y)] :

Substitute (1+3t) for x , (2+t2) for y , and t4 for z in the expression [zln(x+y)] ,

zln(x+y)=(t4)ln[(1+3t)+(2+t2)]=t4ln(t2+3t+3)

Evaluation of line integral Czln(x+y)ds :

Substitute [zln(x+y)] for f(x,y,z) , [t4ln(t2+3t+3)] for f(x(t),y(t),<

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