Laplace transform - step function

[Pi Face]

Homework Statement

f(t)= 1 if 0≤t≤1 ; 0 is t>1
find the laplace transform

Homework Equations

The Attempt at a Solution

I know u(t)= 0 for t<0 and 1 for t≥0

I know I have to shift it and get
u_a(t)=u(t-a)= 1 if 0≤t≤a, 0 if a>1

am I even going the right way?
then I think I integrate it from 0 to inf with
∫e^(-st)u_a(t) dt = ?

not sure what to do from here

 

[vela]

[I know I have to shift it and get
u_a(t)=u(t-a)= 1 if 0≤t≤a, 0 if a>1

What?

 

[Pi Face]

I thought there was a step involving replacing the discontinuous point with a

 

[vela]

There is, but your description doesn't make sense. When is u_a(t) equal to 0 and when is it equal to 1?

 

[Pi Face]

f(t) is 0 when t>1 and 1 when it is between 0 and 1, inclusive.

so for ua(t) wouldn't I just replace the 1 with a?

 

[vela]

I'm not following what you're getting at. Replace what 1 with a?

 

[Pi Face]

You replace the discontinuous point with a right? Which happens to be one in this problem because it has a value of 0 whe greater than 1 and a value of 1 when between 0 and 1. So the discontinuous point would be at 1? Which you replace with a?

 

[vela]

You have it backwards. You don't replace 1 with a. You set a to 1, i.e., u₁(t) = u(t-1). That's the step function shifted to the right by 1.
$$u_1(t) = u(t-1) = \begin{cases}
0 & t<1 \\
1 & t\ge 1
\end{cases}$$