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artofspeed

  • one year ago

integral: are these two expressions equal?

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  1. artofspeed
    • one year ago
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    \[\int\limits_{0}^{a}f(a-x)dx = \int\limits_{}^{}f(0)dx-\int\limits_{}^{}f(a)dx = \int\limits_{a}^{0}f(x)dx\]

  2. klimenkov
    • one year ago
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    You can check this on example. Try \(f(x)=x^2, a=1.\)

  3. klimenkov
    • one year ago
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    It will more easy if you take \(f(x)=1\).

  4. artofspeed
    • one year ago
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    so it this whole expression correct?

  5. sirm3d
    • one year ago
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    only the first and last expressions can be equated.

  6. artofspeed
    • one year ago
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    actually, the first and last aren't equal. But what's wrong in the expression?

  7. klimenkov
    • one year ago
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    \[\int\limits_{0}^{a}f(a-x)dx \ne \int\limits_{}^{}f(0)dx-\int\limits_{}^{}f(a)dx = \int\limits_{a}^{0}f(x)dx\]The primitive of \(f(a-x)\) is \(-\int f(a-x)dx\) and not \(\int f(a-x)dx\). You can check this by taking derivative. Now, I think you can answer your question by yourself.

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