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Math Help - [SOLVED] Integral problem

  1. #1
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    [SOLVED] Integral problem

    If g(x) is continuous for all x, which of the following integrals necessarily hv the same value?

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    I think that 1 and 2 won't have the same value, but I am not completely sure. I have no idea when comparing 1 and 2 to 3 and 4.
    Last edited by solars; July 2nd 2008 at 03:40 PM.
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  2. #2
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    Quote Originally Posted by solars View Post
    If g(x) is continuous for all x, which of the following integrals necessarily hv the same value?

    1.)integral sign (a to b) f(x)dx
    2.)integral sign (a to b) abs(f(x))dx
    3.)integral sign ((a-c) to (b-c)) f(x+c) dx
    4.)integral sign (a to b) (f(x)+c)

    answer choices are:
    A 1, 2 only
    B 1, 3 only
    C 1, 2, 4 only
    D 2,3,4 only
    E no two necessariyt hv the same value

    I think that 1 and 2 won't have the same value, but I am not completely sure. I have no idea when comparing 1 and 2 to 3 and 4.
    For 3. substitute u = x + c. Then:

    \int_{a-c}^{b-c} f(x + c) \, dx = \int_{a}^{b} f(u) \, du which is actually equivalent to \int_{a}^{b} f(x) \, dx since a definite integral does not depend on the (dummy) variable used.

    The correct answer is therefore option B.

    To show that 1. and 2. are not equal, a simple counter-example is sufficient. Consider f(x) = -x, a = 0 and b = 1.

    For 4. note that

    \int_{a}^{b} f(x) + c \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b}c \, dx = \int_{a}^{b} f(x)\, dx + c(b - a) \neq \int_{a}^{b} f(x)\, dx unless a = b or c = 0.
    Last edited by mr fantastic; July 1st 2008 at 11:53 PM.
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