   Chapter 16.2, Problem 6E

Chapter
Section
Textbook Problem

Evaluate the line integral, where C is the given curve.6. ∫C ex dx, C is the arc of the curve x = y3 from (–1, –1) to (1, 1)

To determine

To Evaluate: The line integral C(ex)dx for the arc of the curve x=y3 from the point (1,1) to the point (1,1) .

Explanation

Given data:

The given curve C is the arc of the curve x=y3 from the point (1,1) to the point (1,1) .

Formula used:

Write the expression to evaluate the line integral with respect to arc length.

Cf(x,y)dx=abf(x(t),y(t))x(t)dt (1)

Here,

a is the lower limit of y-coordinate of the curve C and

b is the upper limit of y-coordinate of the curve C .

As the equation of curve is x=y3 , choose y as parameter and write the parametric equations of the curve as follows.

x=y3,y=y

From the given two points, choose y-coordinates as limits to evaluate the integral C(ex)dx .

1y1

Find the expression ex as follows.

Substitute y3 for x in the expression ex ,

ex=ey3

Calculation x(t)dt :

Write the equation of curve as follows.

x=y3

Differentiate on both sides of the expression with respect to y .

ddy(x)=ddy(y3)dxdy=3y2dx=3y2dy

Evaluation of line integral C(ex)dx :

Substitute (ex) for f(x,y) , ey3 for f(x(t),y(t)) , 3y2dy for x(t)dt , (1

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