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I'm trying to check the following system for for time-invariance.

$$ y(t) = \int_0^t x(\lambda) d\lambda$$

Please explain me why it is time-invariant.

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  • $\begingroup$ Could you explain why you think it is time-invariant? $\endgroup$ – Matt L. Apr 14 at 10:47
  • $\begingroup$ I think it's a time-variant... but Is it time invariant? $\endgroup$ – heldi Apr 14 at 11:29
  • $\begingroup$ You should show your calculations and your conclusions. We can't do your homework for you, but we can help you doing it. $\endgroup$ – Matt L. Apr 14 at 11:51
  • $\begingroup$ You better add that to your question. $\endgroup$ – Matt L. Apr 14 at 11:58
  • $\begingroup$ Check how I added the equation to your question (using Latex formatting), and use that to add the steps that you came up with. $\endgroup$ – Matt L. Apr 14 at 12:01
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Note that there are different kinds of (causal) integrators. The one in your question with a fixed finite lower integration limit is not time-invariant, as you've shown yourself (in the comments).

Other variants are

$$y(t)=\int_{-\infty}^tx(\tau)d\tau\tag{1}$$

and

$$y(t)=\int_{t-T}^tx(\tau)d\tau\tag{2}$$

Both of these integrators are time-invariant, which is straightforward to show.

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