   Chapter 16, Problem 6RE

Chapter
Section
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

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

In Exercises 17-20, determine whether the point lies on the graph of the function. 17. (2, 3); g(x)=x21

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

In Problems 19-44, factor completely. 35.

Mathematical Applications for the Management, Life, and Social Sciences

Subtract the following numbers. 23. $127 less$33

Contemporary Mathematics for Business & Consumers

True or False: n=1(1)nnn+3 converges.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

True or False: converges.

Study Guide for Stewart's Multivariable Calculus, 8th

Define the reliability of measurement and explain why and how it is measured.

Research Methods for the Behavioral Sciences (MindTap Course List) 