Definite and Indefinite intregrals.

[QuantumTheory]

Ok.

Im' confused between the difference of definite and indefinite integrals.


$$\int\limits_a_b$$

$$\int$$


The first integral here which is $$\int\limits_a_b$$ is about area below a curve.



Where a and b is the difference of the area under the function f(x). The $$\int\$$ is just the whole of all of the f(x) dx on an area.

Consider we have an area under the curve.

We will call the function f(x) = $$x^2$$

The area under the curve is then defined as:

$$\int\limits_a_b f(x) dx = dL$$

The $$\int\limits_a_b$$ is defined as all of dx of the function f(x) from a to b.

dx is a small infinitely small piece of the area under the curve.

dL is defined as the area.

I do not understand the integral:


$$\int$$ , which has no limits (a to b).

I know that this integral is backwards differenatation and requires a constant (I don't know what "arbitary" means, I think it means "fixed"?)

Such that,

$$\int x^2$$ = $$1/3^2 + C$$


Help please?

Thanks

 

[QuantumTheory]

OH!

I think it requires a constant because there is no limit! (like the area integral)
(No a to b)

Just popped in my head..am I right?

 

[mathman]

The indefinite integral has the variable as part of the answer. the integral of x² is x³/3 + c. (You can verify by taking the derivative with respect to x - the derivative of c is 0)

 

[courtrigrad]

The indefinite integral allows the upper or lower limit of integration to vary, while definite integration both the upper and lower limits are fixed. Hence we have the fundamental theorem of calculus.

Hopw this helps

 

[dextercioby]

The indefinite integral of a function (or the primitive) is nothing but the function you need to differentiate as to obtain the original function (a.k.a.the integrand).
The definite integral has the geometrical interpretation of the area under the graphic of the function which constitutes the integrand,and the Leibniz-Newton theorem establishes the connection between the two notions.
This is the simplest explanation one can give.