Definite and Indefinite intregrals.

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In summary, the difference between definite and indefinite integrals lies in the presence or absence of limits of integration. The definite integral represents the area under a curve, while the indefinite integral is the function that, when differentiated, gives the original function. The fundamental theorem of calculus connects these two concepts.
  • #1
QuantumTheory
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Ok.

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


[tex]\int\limits_a_b[/tex]

[tex]\int[/tex]


The first integral here which is [tex]\int\limits_a_b[/tex] is about area below a curve.



Where a and b is the difference of the area under the function f(x). The [tex]\int\[/tex] 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) = [tex]x^2[/tex]

The area under the curve is then defined as:

[tex]\int\limits_a_b f(x) dx = dL[/tex]

The [tex]\int\limits_a_b[/tex] 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:


[tex]\int[/tex] , 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,

[tex]\int x^2[/tex] = [tex]1/3^2 + C[/tex]


Help please?

Thanks :wink:
 
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  • #2
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?
 
  • #3
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)
 
  • #4
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
 
  • #5
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.
 

1. What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration, while an indefinite integral does not. In other words, a definite integral gives a numerical value, while an indefinite integral gives a function.

2. How do you solve a definite integral?

To solve a definite integral, you must first find the antiderivative of the function. Then, plug in the upper and lower limits of integration and subtract the results to find the numerical value.

3. What is the purpose of indefinite integrals?

Indefinite integrals are used to find the general solution of a differential equation. They can also be used to find the equation of a curve given its derivative.

4. What is the relationship between derivatives and definite integrals?

Definite integrals and derivatives are inverse operations of each other. The derivative of a function is the slope of the tangent line at a specific point, while the definite integral is the area under the curve between two points.

5. How do you know when to use definite or indefinite integrals?

Definite integrals are used when you want to find the numerical value of the area under a curve between specific limits. Indefinite integrals are used when you want to find the general solution of a differential equation or the equation of a curve given its derivative.

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