Isn't indefinite integral of a curve just the formula for the area under that curve?

Question

When you put the upper and lower bounds, it gives area between two co-ordinates. So isn't it just like the formula for area of a rectangle? You put the values of length and breadth and get the area.

Answer 1

Thanks to the Fundamental Theorem of Calculus, the indefinite integral (antiderivative) of a continuous function is indeed related to its definite integrals. However, the concepts are not identical. Functions that are not continuous generally don't have antiderivatives, but they often have definite integrals.

Answer 2

The indefinite integral really describes a family of functions, all of which have the same derivative, since the "indefinite form" is $F(x) = F(a) + \int_a^x f(x)dx$ where $f=F'$ and $F(a)$ is arbitrary. Since any $F(a)$ is consistent with $F'(x) = f(x)$, this formula is talking about an entire family of objects, each with potentially a different $F(a)$.

On the other hand, the definite integral is the area under the curve, like you said, so $F(b)-F(a) = \int_a^b f(x)dx$, which is not arbitrary and is the same for every member of the family.

Answer 3

Precisely not. The indefinite integral does not specify the bounds so the "length" is unknown. And so is the starting abscissa. Both are required to define an area.

Also think that the indefinite integral includes an arbitrary constant, and has no specific value.

Answer 4

An indefinite integral is not a formula you plug two numbers into. It is literally the anti-derivative, that is, if you take the derivative of a function $f$ you get a new function $f'$, and if you then take the antiderivative of $f'$ you get back $f$ plus some constant.

For convenience so we don’t have to define too many symbols, we re-use the symbol $\int$ by placing numbers above and below it to denote a definite integral.

But what is the derivative of $\int_1^3 x\,dx$? What is the derivative of $\int_a^b x\,dx$?

It’s true that there is a relationship between an integral (of either kind) and the formula for the area of a rectangle. You need the rectangle formula to define integration, and conversely integration can be a tool for computing an area. But alarm bells go off in my head when I hear “isn’t it just ... .” If that’s the only way you understand an integral then you may run into trouble as soon as a slightly different application comes along.