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Math Help - Upper and lower bounds of integration.

  1. #1
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    Upper and lower bounds of integration.

    Suppose that we know that 1 is greater than or equal to g(x)dx with bounds 0 to 4 and is equal or greater to 8. Determine the best possible upper and lower bound for x g(x^2)dx with bounds 0 to 2 that can be deduced from this information.
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  2. #2
    Moo
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    Hello,

    Quote Originally Posted by Hibijibi View Post
    Suppose that we know that 1 is greater than or equal to g(x)dx with bounds 0 to 4 and is equal or greater to 8. Determine the best possible upper and lower bound for x g(x^2)dx with bounds 0 to 2 that can be deduced from this information.
    Do a substitution t=x
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Hibijibi View Post
    Suppose that we know that 1 is greater than or equal to g(x)dx with bounds 0 to 4 and is equal or greater to 8. Determine the best possible upper and lower bound for x g(x^2)dx with bounds 0 to 2 that can be deduced from this information.
    This is just a glorified integration by substitution problem.

    Consider \int_0^2 x g(x^2)~dx

    Let u = x^2

    \Rightarrow du = 2x~dx

    \Rightarrow \frac 12 ~du = x~dx

    Changing the limits we find:

    when x = 2, u = 2^2 = 4
    when x = 0, u = 0^2 = 0

    So our integral becomes:

    \frac 12 \int_0^4 g(u)~du

    I leave the easy part to you


    EDIT: Well thanks for making me type all this, Moo
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