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

hanhao

hello maths experts
is the following true?


graphically, this is how i view it

LeonhardEuler

Yes, that's pretty much correct, but the right hand side is missing a "+C" because it is an indefinite integral.

hanhao

how about this?

http://img108.imageshack.us/img108/1626/38jk1.jpg
is u(x-a) redundant? can i remove it like this?

LeonhardEuler

No, the integral is constant for x<a. The u(x-a) keeps the part of the integral that is dependant 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.

hanhao

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)] ??

LeonhardEuler

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 in to the function you proposed:
I=[F(c)-F(a)]-[F(b)-F(a)]=F(c)-F(b)
which is not necessarily zero.

nightworrier

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 ?