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Math Help - Double Integration

  1. #1
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    Double Integration

    ∫R (x^1/2 - y^1/2) dydx
    where y=x^2, y= x^1/4, x=0, x=1
    im struggling to solve this!! i came up with an answer of 1.5 but i dont think i have done it correct!...could anyone help please?
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  2. #2
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    \int_{0}^{1}\int_{x^{\frac{1}{4}}}^{x^{2}}(\sqrt{x  }-\sqrt{y})dydx

    First, integrate wrt y:

    \int_{x^{\frac{1}{4}}}^{x^{2}}{\sqrt{x}-\sqrt{y}}dy=y\sqrt{x}-\frac{2y^{\frac{3}{2}}}{3}

    Now use the limits of integration for y. By the FTC:

    \frac{-2x^{3}}{3}+x^{\frac{5}{2}}-x^{\frac{3}{4}}+\frac{2x^{\frac{3}{8}}}{3}

    Now, integrate this wrt x:

    \int_{0}^{1}{\left[\frac{-2x^{3}}{3}+x^{\frac{5}{2}}-x^{\frac{3}{4}}+\frac{2x^{\frac{3}{8}}}{3}\right]}dx

    We get:

    \frac{x^{4}}{6}+\frac{2x^{\frac{7}{2}}}{7}-\frac{4x^{\frac{7}{4}}}{7}+\frac{16x^{\frac{11}{8}  }}{33}

    Now, use the limits of integration for x, 0 and 1:

    We get \frac{154}{5}
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