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Math Help - Order Of Integration Switch

  1. #1
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    Order Of Integration Switch

    I've been given the double integral \int_{x= 0}^{1}\int_{y=x}^{3x}xy dydx and instructed to change the order of integration and solve. After solving the given integrals I found they equaled 1.

    I reasoned that the initial graph is bounded by two lines y=x and y=3x and bounded on the x-axis between x=0 and x=1; this formed a skewed sort of triangle.

    Here's what I changed it to: \int_{y= 0}^{3}\int_{x=(1/3)y}^{x}xy dxdy

    I either I'm I'm messing up or my professor's solution is wrong...help?
    Last edited by spruancejr; November 6th 2011 at 01:44 PM.
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  2. #2
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    Re: Order Of Integration Switch

    Did you just change around all the x's and y's? You must actually look at the geometry.


    First Piece \int_{0}^{1}\int_{y/3}^{y} xy\;dxdy


    Second Piece \int_{1}^{3}\int_{y/3}^{1} xy\;dxdy
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  3. #3
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    Re: Order Of Integration Switch

    When I solve the two integrals, I get 9 as opposed to 1 which was the answer prior to switching the order of integration. Should the two answers be the same?
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    Re: Order Of Integration Switch

    Quote Originally Posted by spruancejr View Post
    When I solve the two integrals, I get 9 as opposed to 1 which was the answer prior to switching the order of integration. Should the two answers be the same?
    Check your upper bound on the second set of integrals.
    \int_1^3 {\int_{y/3}^1 {xydxdy} }
    Last edited by Plato; November 6th 2011 at 03:54 PM.
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  5. #5
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    Re: Order Of Integration Switch

    I'm confused by the second integral going from 0 to 1. Shouldn't it be from 1 to 3 since the second (top triangle) is bounded between 1 and 3 in the y direction?
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  6. #6
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    Re: Order Of Integration Switch

    Quote Originally Posted by spruancejr View Post
    I'm confused by the second integral going from 0 to 1. Shouldn't it be from 1 to 3 since the second (top triangle) is bounded between 1 and 3 in the y direction?
    CORRECTED NOW.
    That was a copy&paste error.
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