   Chapter 16.2, Problem 14E

Chapter
Section
Textbook Problem

Evaluate the line integral, where C is the given curve.14. ∫C y dx + z dy + x dz, C: x = t , y = t, z = t2, 1 ⩽ t ⩽ 4

To determine

To Evaluate: The line integral C(y)dx+(z)dy+(x)dz for a curve.

Explanation

Given data:

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

C:x=t,y=t,z=t2,1t4

Calculation of dx :

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

ddt(x)=ddt(t)dxdt=12tdx=12tdt

Calculation of dy :

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

ddt(y)=ddt(t)dydt=1dy=dt

Calculation of dz :

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

ddt(z)=ddt(t2)dzdt=2tdz=2tdt

Evaluation of line integral C(y)dx+(z)dy+(x)dz :

Substitute t for x , t for y , t2 for z , 12t for dx , dt for dy , 2tdt for dz , 1 for lower limit, and 4 for upper limit in the line integral C(y)dx+(z)dy+(x)dz ,

C(y)dx+(z)dy+(x)dz=14(t)12tdt+(t2)dt+(t)2tdt=14(12t12+t2+2t32

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