   Chapter 16.2, Problem 13E

Chapter
Section
Textbook Problem

Evaluate the line integral, where C is the given curve.13. ∫C xyeyz dy, C: x = t, y = t2, z = t3, 0 ⩽ t ⩽ 1

To determine

To Evaluate: The line integral C(xyeyz)dy for a curve.

Explanation

Given data:

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

C:x=t,y=t2,z=t3,0t1

Calculation of expression (xyeyz) :

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

xyeyz=(t)(t2)e(t2)(t3)=t3et5

Calculation of dy :

Differentiate on both sides of the expression y=t2 with respect to t as follows.

ddt(y)=ddt(t2)dydt=2tdy=2tdt

Evaluation of line integral C(xyeyz)dy :

Substitute (t3et5) for (xyeyz) , 2tdt for dy , 0 for lower limit, and 1 for upper limit in the line integral C(xyeyz)dy ,

C(xyeyz)dyt=01(t3et5)

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