What's the meaning of computing an integral at a given point?

Question

Let $f$ be a function. If one finds $\displaystyle \frac{\mathrm d}{\mathrm dx}f$ and computes it at $x=a$, then one gets the rate of change of $f$ at $a$. That can be useful in some situations. But if one finds $\int f \space \mathrm dx$ and computes it at $x=a$, what information can we obtain about the function $f$?

Answer 1

You use the indefinite-integral tag, so I'll assume you are talking about indefinite integrals. In that case, recall that when integrating an integrable function f(x): $\displaystyle \int f \,dx$, we obtain a family of functions: $F(x) + C$, where $C$ can be any constant.

So we can not evaluate $F(x) + C$ at $a$ and obtain a distinct value for $a$, without knowing $C$.

Answer 2

Because antiderivatives are only unique up to a constant, the answer is: not much, since evaluating the antiderivative can, in fact, give you any number (see @amWhy's answer).

However, we could reinterpret your question in the following way:

Given a function $f(x)$, an antiderivative is a function $F(x)$ such that $F^\prime(x)=f(x)$. In other words, $F(x)$ is a function which solves the equation

$$ F^\prime(x)=f(x) $$

As previously mentioned, this equation typically has infinitely many solutions - all varying by a constant. For example, consider

$$ F^\prime(x)=e^x $$ What type of function has derivative $e^x$? Well, it must be the exponential function plus a constant:

$$ F(x)=e^x+C $$ How might we restrict this collection of solutions? Well, suppose we knew the value of $F(x)$ at a point - say we know $F(0)=1$. Then, as it turns out, we only get one solution to this equation - we choose the constant of integration so that $F(0)=1$. In our example above, we would choose $C=0$ since $e^0=1$. This type of process is just the beginning of the (vast and fascinating) study of Differential Equations - determining a function from knowledge of its derivatives.

So what is the meaning of, say, $F(1)$ in the example above? It would be the value of a function whose derivative is $e^x$, and whose value at zero is $1$.