What does area of definite integral represent?

It is the area between the curve and the x-axis, bounded on the left and right by the limits of integration.  With a definite integral, you don't need to worry about a constant of integration.

a. Definite integrals are used to compute the area under curves from point a to point b.

b. As for predicting a value of an antiderviative, no it doesn't work that way.

integrate x from a to b where n is a continuous function
y=(x^2)/2 from 1 to 3 is (1/2) (9-1)=4

dy/dx= x where Y(0)=-2
y=x^2/2+c
-2=0+c
y=x^2/2-2

you may be confusing integrating and solving for a constant at a particular solution

The area calculated using Integral Calculus can represent a number of interpretations. The relationship between the variables will be an indication of what the calculated area will be called.

In applied Calculus, for example, if your coordinate system represents, say, Pressure/Volume relationship, the area under the curve within the boundaries defined by the limits will represent "Energy" or "Work" done.

If your variables are Force and Distance, then the area under the curve is also called "Energy" and "Work" done.

Another example is that if your coordinate system represents, say, velocity/time relationship, the area under the curve within the boundaries defined by the limits will represent the total Distance travelled by a particle or a body.

There are so many examples of what the area under the curve will represent. It all depends on how the variables are related.

And, NO -- the area under the curve will not predict any value of the antiderivative function. If you integrate any function within two limits, you will always get a constant -- nothing more, nothing less. It will not mean anything except that it is the area under the curve.