Work Kinetic Energy Theorem

1.)

A net force F = -2.00x i - 3.00 y j acts on a certain object with a mass m=2.00 kg while it moves from a point with coordinates (2.00, 3.00) to a point with coordinates (2.00, -1.00) along a straight line. At the first point, the speed of the object was 2.00 m/s. Using the work kinetic energy.

a.) find the speed of the object when it is at second point
b.) Find the speed of the object at the second point in this case.


This is the question, and this is the answer from my professor:

Ans:

W = Kinetic Energy(final) - Kinetic Energy(initial)

Integral( F dot product dr) = Kinetic Energy(final) - Kinetic Energy(initial)

r = 2 i + y j
dr = 0 + dy j

W = Integral ( (-2x i - 3y j) dot product ( 0x i + dy j) )
= Integral ( 0 + (-3y) dy )
= Integral from -1 to 3 ( -3y dy )

= 12 J

12 J = 1/2 (2)( vf ^2 )- 1/2 (2) (2)^2
16 J = vf ^2

vf = 4 J



My question is, how can I get the r and the dr ? And why do we need to useIntegral( F dot product dr) to find work ( is that because the force is varying? but how? )

Please explain to me. really need help