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Don't understand this application of the transform of an integral

  1. Oct 2, 2013 #1
    1. The problem statement, all variables and given/known data

    $$L\{ 1*{ t }^{ 3 }\}$$

    2. Relevant equations

    $$L\{ \int _{ 0 }^{ t }{ f(\tau )d\tau } \} =\frac { L\{ f(t)\} }{ s }$$

    3. The attempt at a solution

    $$L\{ 1*{ t }^{ 3 }\} \\ =L\{ \int _{ 0 }^{ t }{ { (t-\tau ) }^{ 3 }d\tau } \} \\ =\frac { L\{ { t }^{ 3 }\} }{ s } \\ =\frac { 6 }{ { s }^{ 5 } }$$

    The function in the integral is a function of t and tau so how was the identity under "relevant equations" applied?
     
  2. jcsd
  3. Oct 2, 2013 #2
    Yes, you have ##\int_0^tf(\tau,t)\,d\tau## in the integral instead of ##\int_0^tf(\tau)\,d\tau## and the formula for integral doesn't work. This is convolution so:

    ##L\{1*t^3\}=L\{1\}\cdot L\{t^3\}##
     
  4. Oct 2, 2013 #3

    vela

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    Use the substitution ##u=t-\tau## on the integral to convert it to a more familiar form.
     
  5. Oct 3, 2013 #4
    @szynkasz: Thanks. I solved it.

    knawF7X.png

    Any suggestions?
     
  6. Oct 3, 2013 #5

    vela

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    Remember that ##u## is a dummy variable.
     
  7. Oct 3, 2013 #6
    Yeah, I substituted it back in the last step so there is no more u. I'm not sure how that would help me.
     
  8. Oct 3, 2013 #7

    vela

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    I don't think you understood my point. In the integral ##\int_0^t u^3\,du##, it doesn't matter what letter we use for the variable of integration. It's a dummy variable.

    http://mathworld.wolfram.com/DummyVariable.html

    If you evaluated the integral, what variable is it a function of?
     
  9. Oct 3, 2013 #8
    It's a function of u which is the same thing as saying it's a function of t - tau.
     
  10. Oct 3, 2013 #9

    vela

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    Really? Did you actually calculate ##\int_0^t u^3\,du##?
     
  11. Oct 3, 2013 #10
    No, I didn't. The textbook says to apply the identity under "2. Relevant equations" first.
     
  12. Oct 3, 2013 #11

    vela

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    You wrote
    What variable is ##f## a function of on the left-hand side? What variable is ##f## a function of on the right-hand side? Are they the same? Why not?
     
  13. Oct 3, 2013 #12
    On the left hand side, f is a function of tau. On the right hand side, f is a function of t. They are not the same. I'm not sure why they're not the same but I can dig up the proof for it I guess. I think they're not the same because we're doing a transformation.
     
    Last edited: Oct 3, 2013
  14. Oct 3, 2013 #13
    I found the proof:

    jhaaAgt.png

    Honestly, that proof (as most proofs) went right over my head.
     
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