Question

2. (8 points) Let F- try - vy2. Evaluate the line integralF dl using the f ollowing paths: a) (2 points) Going from 0, 0 to r 2,y 1 along the straight line y b) (2 points) Going from 0, 0 to r 2,y-1 along the parabola y c) (2 points) Going from 0,y 0 to 2,y 1 by going first along a vertical line from0,y0 to x-0, y-l then along the horizontal line x , y-i to x-2, y-1, d) (2 points) Going from 0,y to 2,y 1 ing the path (z(t), y()) where () 2t3,y)

Answer 1

Given,

.................................(1)

and,

or,   

or, ....................................(2)

(a) Given,

or,   

And the range of x is from x= 0 to x = 2 .

Putting all the values in equation (2), we get

or,   

or,

or,   

or,

(b)  Given,

  

so,   

and the range of x is from x = 0 to x = 2.

Putting all the values in equation (2), we get

  

or,

or,

16 32 6 5

or,   

or,   

(c)  In this case, the integration is along the two paths : first along a vertical line from y = 0 (x= 0) to y = 1(x = 0) (i.e. along y - axis) and then along a horizontal line from x= 0 (y = 1) to x = 2 (y = 1) (i.e. parallel to the x - axis).

Let's denote the first path (i.e. along the y - axis) by 'A' and the second path (i.e. parallel to x -axis) by 'B' .

Integral along the path A :

The equation of the line (path of integration) will be , x = 0.

So, dx = 0

So, the integration will be

or,

or,

or,

Integral along the path B :

The equation of the line(path of integration) will be : y = 1 .

So, dy = 0

So, the integration will be

or,

or,

or,  

Hence the final result will be

or,

(d)  Given,

   and

So,    and
dy = 2tdt

And, for x = 0 ; t = 0 and for x = 2 ; t = 1 this condition (or limits) is also satisfied for y = 0 and y =1 .

So,

or,   .

or,   

or,

or,

or,

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