Question

The work done by the Force F( x, y) = (-y, x) in moving a particle along the boundary of the ellipse 9x2 + 4 - y² = 36 is 6 ? justify your answer

Answer 1

False.  The total work done during the traversal of the particle along the ellipse is 12 π units.
======

Vector:  F(x,y) = -y i + x j ,   i, j are unit vectors.

Ellipse :  ABCD:     A(0,2), B(0,3), C(-2,0), D(0,-3)
               9 x² + 4 y² = 36   or,   x²/2²  +  y²/3² = 1
               a = 2, b = 3 

The particle moves along the path of the Ellipse: 
     vector s = x i + y j         ds = dx i + dy j

Let x = 2 cosФ,    dx = - 2 sin Ф dФ
  From A to B to C, x varies from 2 to 0 to -2.  Ф varies from 0 to π/2 to π.
  From C to D to A, x varies from -2 to 0 to 2. Ф varies from π to 3π/2 to 2π.

Let y = 3 sinФ,    dy = 3 cosФ dФ
   From A to B to C, y varies from 0 to 3 to 0.  (Ф varies from 0 to π/2 to π)
   From C to D to A, y varies from 0 to -3 to 0. (Ф varies from π to 3π/2 to 2π).

Work = F(x,y) . ds = dot product
    dW = - y dx + x dy  = 6 sin² Ф dФ + 6 cos²Ф dФ
          = 6 dФ

Work done:  Integral of  6 dФ with Ф from 0 to 2π 
         = 12 π units.