What does indefinite integral give us?

Question

If we integrate a function without puting limits we get a function only, and if with limits we get a constant value. The in the 1st case arent we just changing the function?

Answer 1

A headache!

Answer 2

If you use the limits, you will get the area enclosed.
If you dont then you will get the equation of the curve that encloses that area.

Indefinate :
In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i.e., F′ = f. The process of solving for antiderivatives is antidifferentiation (or indefinite integration). Finding an expression for an antiderivative is harder than calculating a derivative, and may not always be possible. Antiderivatives are related to integrals through the fundamental theorem of calculus, and provide a convenient means for calculating the integrals of many functions.

More details :http://en.wikipedia.org/wiki/Indefinite_integral

Definate Integral :
In calculus, the integral of a function is a generalization of the concept of a sum, typically with direct physical interpretation pertaining to area, mass, or volume. The process of finding integrals is integration, in its mathematical meaning. Unlike the closely-related process of differentiation, there are several possible definitions of integration, with different technical underpinnings. They are, however, compatible; any two different ways of integrating a function will give the same result when they are both defined.

The word "integral" may also refer to antiderivatives in a mild abuse of language. (The antiderivative of a function is that function whose derivative is equal to the first function.) Though they are closely related through the fundamental theorem of calculus, the two notions are conceptually distinct. When one wants to clarify this distinction, an antiderivative integral is referred to as an indefinite integral (a function), while the integrals discussed in this article are termed definite integrals.

More details : http://en.wikipedia.org/wiki/Definite_integral

Hope you understand this.

Answer 3

No. In the first case, we get a function that can theoretically be applied throughout the whole domain.

For example, integrate y=x, you get y=x^2/2, This applies from
-infinity to infinity

But if you applied bounds of 0 to 2, you'd get y = 2-0 = 2. That means you've evaluated the integral only between those two points. It's the second case that actually limits your options.