Question

Evaluate the line integral, where C is the given curve.

C

xe^yzds, Cis the line segment from

(0, 0, 0) to (2, 3, 4)

Answer 1

To fine the line integral of a curve using their parameterized equivalents, we use the formula

To find this,let'S start by parameterizing the curvein vector form:

Our points are (1,2,3) and (0,0,0)

Direction vectors <a,b,c> = <x1-x0, y1-y0, z1-z0>

<a,b,c>= <1-0,2-0,3-0> =<1,2,3>

Parametric equations are of the form x=m+nt,where m denotes corresponding cordinate and n denotes corresponding direction vectors

Hence substitute (x0,y0,z0) as (0,0,0) and <a,b,c> as <1,2,3>

Therefore x=0+(1)t y=0+(2)t z=0+(3)t

This gives x=t y=2t z=3t

And we always consider the limits of scalar parameter t are 0 to 1. 0<=t <=1

Now we have parametric equation r(t)=<t, 2t, 3t> ,0<=t <=1

Taking the derivative of curve r'(t)= <1,2,3>

Calculating magnitude of derivative=

Substituting values for x,y,z and ds with parameterized eqns using the formula

  

Put u=6t2  ,Then du=12t.dt limit 0 changes to 0 ,and 1 changes to 6

using in equation  

This is equal to   

Therefore the line integral of the curve is or to be precise ,by substituting values, 125.479