The definite integral of f(x) is a NUMBER and represents the area under the curve f(x) from x=a to x=b.
The indefinite integral of f(x) is a FUNCTION and answers the question, "What function when differentiated gives f(x)?"
The FTC relates these two integrals in the following manner:
To compute a definite integral, find the antiderivative (indefinite
integral) of the function and evaluate at the endpoints x=a and x=b.