Learning Integration with unit step function like u(x - a)

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Discussion Overview

The discussion revolves around the integration of functions involving the unit step function, specifically u(x - a). Participants explore the implications of using the unit step function in integrals, its graphical representation, and the conditions under which it may be considered redundant.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the correctness of an integral involving the unit step function and seeks confirmation from others.
  • Another participant notes that the right-hand side of the integral is missing a constant of integration, "+C," indicating an understanding of indefinite integrals.
  • A participant asks whether the unit step function u(x - a) can be removed from an integral, suggesting that it may be redundant.
  • In response, it is argued that the unit step function is necessary to maintain the integral's behavior for values of x less than a, as it keeps the integral constant in that region.
  • One participant seeks a graphical understanding of how the function u(x - a)[F(x) - F(a)] compares to [F(x) - F(a)], questioning the representation of these functions in a provided graph.
  • Another participant clarifies that the bottom graph does not represent [F(x) - F(a)], but rather f(x)u(x - a), and explains the implications of integrating this function over specified limits.
  • A participant raises a question about computing integrals with limits from negative to positive infinity when multiplied by a unit step function, indicating a need for clarification on this process.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of the unit step function in certain integrals, and there is no consensus on whether it can be considered redundant. The discussion remains unresolved regarding the graphical interpretations and the computation of integrals involving the unit step function.

Contextual Notes

Some participants' claims depend on specific assumptions about the behavior of functions at certain limits, and the discussion includes unresolved mathematical steps related to the integration process with the unit step function.

hanhao
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hello maths experts
is the following true?
http://img9.imageshack.us/img9/4596/int15oe.jpg

graphically, this is how i view it
http://img9.imageshack.us/img9/179/int28ut.jpg
 
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Yes, that's pretty much correct, but the right hand side is missing a "+C" because it is an indefinite integral.
 
how about this?

http://img108.imageshack.us/img108/1626/38jk1.jpg
is u(x-a) redundant? can i remove it like this?
 
Last edited by a moderator:
hanhao said:
how about this?

http://img108.imageshack.us/img108/1626/38jk1.jpg
is u(x-a) redundant? can i remove it like this?
No, the integral is constant for x<a. The u(x-a) keeps the part of the integral that is dependent on x zero for x<a, so it is just the constant of integration before that. The integral should be u(x-a)[F(x)-F(a)]+C.
 
Last edited by a moderator:
i want to understand this graphically
how would the graph of u(x-a)[F(x)-F(a)] look like compared to [F(x)-F(a)] ??
am i correct to say that my bottom graph is [F(x)-F(a)] ??
 
I am assuming you mean the bottom graph in this image, so tell me if I am wrong:
http://img9.imageshack.us/img9/179/int28ut.jpg
This is not the graph of [F(x)-F(a)]. The function itself is f(x)u(x-a). The area represents the integral of this, which is u(x)[F(b)-F(a)], where b is the upper limit.

[F(x)-F(a)] represents an antiderivative of f(x) without the step function. Suppose b and c are both less than a. Obviously the integral,I, of f(x)u(x-a) from b to c is zero, but look what happens when you plug this into the function you proposed:
I=[F(c)-F(a)]-[F(b)-F(a)]=F(c)-F(b)
which is not necessarily zero.
 
Last edited by a moderator:
We have limits between infinite to minus infinite how can i compute when i multiply a function with an unit step function. i mean i have an integral the limits of that integral is infinite to minus infinite and inside the integral i have f(t).u(t-a) this. So how can i compute this integral ?
 

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