In the following document( page 6/7) : http://www.math.ubc.ca/~malabika/teaching/ubc/spring12/math105/sec204/LectureNotes-Feb6.pdf
the area function of a function f, denoted by A, and defined as
the integral of f(t)dt from a to x
is determined as " definite integral" ( and also as a " connection" between indefinite and definite integral).
This seems confusing to me because I thought that the definite integral ( as such) was not a function but a number; while the area function is surely a function,, namely , the function that maps every x to the area between the vertical line passing through a, the vertical line passsing through x, the X axis and the curve representing f.
In other words, I think that the area function maps every x to the indefinite integral ( up to x).
Hence my question : a) what is the link between the indefinite integral and the area function? b) in which way does the area function provide a connection between the indefinite integral and the definite one?
Best Answer
The definite integral is a number if the upper and lower bounds are both numbers. Since x is a variable, the Area function changes depending on the value of x. You will get numerical values for values of x.