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Math Help - integration question

  1. #1
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    integration question

    Hi,

    I'm stuck on the domain of integration change in the following problem:

    I have a Uniform distribution, U(a,b).

    I'm deriving the variance:

    var = \int^b_a \left(x-\frac{b+a}{2}\right)^2 \frac{1}{b-a} dx

    Use a change of variable substitution: y =x - \frac{b+a}{2}

    Take the constant out

    var = \frac{1}{b-a} \int^\frac{b-a}{2}_{-\frac{b-a}{2}} y^2 dy

    I'm not sure why the domain of integration has changed from a to b, to \frac{b-a}{2}. I'm presuming it's to do with the y change of variable. Any ideas?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by notgoodatmath View Post
    Hi,

    I'm stuck on the domain of integration change in the following problem:

    I have a Uniform distribution, U(a,b).

    I'm deriving the variance:

    var = \int^b_a \left(x-\frac{b+a}{2}\right)^2 \frac{1}{b-a} dx

    Use a change of variable substitution: y =x - \frac{b+a}{2}

    Take the constant out

    var = \frac{1}{b-a} \int^\frac{b-a}{2}_{-\frac{b-a}{2}} y^2 dy

    I'm not sure why the domain of integration has changed from a to b, to \frac{b-a}{2}. I'm presuming it's to do with the y change of variable. Any ideas?
    Remember your limits were originally in terms of x.

    If y(x)=x-\frac{b+a}{2}, what is y(a) and y(b)? These will give you the new limits of integration with respect to y.
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  3. #3
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    Quote Originally Posted by notgoodatmath View Post
    Hi,

    I'm stuck on the domain of integration change in the following problem:

    I have a Uniform distribution, U(a,b).

    I'm deriving the variance:

    var = \int^b_a \left(x-\frac{b+a}{2}\right)^2 \frac{1}{b-a} dx

    Use a change of variable substitution: y =x - \frac{b+a}{2}

    Take the constant out

    var = \frac{1}{b-a} \int^\frac{b-a}{2}_{-\frac{b-a}{2}} y^2 dy

    I'm not sure why the domain of integration has changed from a to b, to \frac{b-a}{2}. I'm presuming it's to do with the y change of variable. Any ideas?
    When you change variables you have to plug the limits of integration into the expression that you use to change variables, so just plug in a for x in y =x - \frac{b+a}{2} and also plug in b to get your lower and upper limit
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  4. #4
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    got it, thanks
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