is the area function of a function f a definite integral or an indefinite integral of function f?

Question

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?

Answer 1

Let me fix the domain of discourse by explicitly fixing a continuous function $f : I \to \mathbb R$ and its domain $I \subset \mathbb R$, where $I$ is a subinterval of $I$.

Next, let me fix two numbers $a < b$ in the interval $I$.

Having fixed all of those choices, then yes, the definite integral $\int_a^b f(t) dt$ is indeed a single number.

But now what one can do is to replace the fixed number $b$ by a variable $x$, whose values vary over the interval $I$. One then defines a function $$F(x) = \int_a^x f(t) dt $$ (if $a<x$ this works just fine but otherwise we must be a bit more careful: if $a=x$ then $F(x)=0$; and if $a<x$ then $F(x) = -\int_x^a f(t) dt$).

And now we can state:

The Fundamental Theorem of Calculus: $F(x)$ is an antiderivative of $f(x)$, meaning that $F'(x) = f(x)$ for all $x \in I$.

The word indefinite integral is nothing more or less than a synonym for "antiderivative", and so we can also say that $F(x)$ is an indefinite integral of $f(x)$.

Answer 2

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.