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Math Help - hard triple integral

  1. #1
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    hard triple integral

    Hey everyone, I have a really hard triple integral that I need help on.

    \int_2^5 \ \int_0^{\sqrt{16-x^2}} \ \int_0^{\frac{x^2+y^2}{4}} \ \frac{2z}{\sqrt{x^2+y^2}} \ \ dzdydx


    My working:

    \int_2^5 \ \int_0^{\sqrt{16-x^2}} \  [ \frac{z^2}{\sqrt{x^2+y^2}}]_{z=0}^{z=\frac{x^2+y^2}{4}} \ \ dydx

    \int_2^5 \ \int_0^{\sqrt{16-x^2}} \  \frac{(x^2+y^2)^\frac{3}{2}}{16} \ \ dydx

    But then to integrate with respect to y is very tricky for me, I can get it by calc but it is big and I want to know how to do it by hand.

    Any help would be nice.

    Thanks
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  2. #2
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    I suspect converting to polars is the easiest way, because otherwise you get a VERY long and tedious integral...


    Let y = x\sinh{t} so that dy = x\cosh{t}\,dt and the first integral becomes:

    \int{\frac{(x^2 + y^2)^{\frac{3}{2}}}{16}\,dy} = \int{\frac{[x^2 + (x\sinh{t})^2]^{\frac{3}{2}}}{16}\,x\cosh{t}\,dt}

     = \int{\frac{(x^2 + x^2\sinh^2{t})^{\frac{3}{2}}}{16}\,x\cosh{t}\,dt}

     = \int{\frac{[x^2(1 + \sinh^2{t})]^{\frac{3}{2}}}{16}\,x\cosh{t}\,dt}

     = \int{\frac{(x^2\cosh^2{t})^{\frac{3}{2}}}{16}\,x\c  osh{t}\,dt}

     = \int{\frac{(x\cosh{t})^3}{16}\,x\cosh{t}\,dt}

     = \int{\frac{x^4\cosh^4{t}}{16}\,dt}

     = \frac{x^4}{16}\int{\cosh^4{t}\,dt}

     = \frac{x^4}{16}\int{(\cosh^2{t})^2\,dt}

     = \frac{x^4}{16}\int{\left(\frac{1}{2}\cosh{2t} + \frac{1}{2} \right)^2\,dt}

     = \frac{x^4}{16}\int{\frac{1}{4}\cosh^2{2t} + \frac{1}{2}\cosh{2t} + \frac{1}{4}\,dt}

     = \frac{x^4}{16}\int{\frac{1}{4}\left(\frac{1}{2}\co  sh{4t} + \frac{1}{2}\right) + \frac{1}{2}\cosh{2t} + \frac{1}{4}\,dt}

     = \frac{x^4}{16}\int{\frac{1}{8}\cosh{4t} + \frac{1}{8} + \frac{1}{2}\cosh{2t} + \frac{1}{4}\,dt}

     = \frac{x^4}{16}\int{\frac{1}{8}\cosh{4t} + \frac{1}{2}\cosh{2t} + \frac{3}{8}\,dt}

     = \frac{x^4}{16}\left(\frac{1}{32}\sinh{4t} + \frac{1}{4}\sinh{2t} + \frac{3}{8}t\right) + C

     = \frac{x^4}{16}\left(\frac{1}{16}\cosh{2t}\sinh{2t} + \frac{1}{2}\cosh{t}\sinh{t} + \frac{3}{8}t\right) + C

     = \frac{x^4}{16}\left[\frac{1}{8}(2\sinh^2{t} + 1)\cosh{t}\sinh{t}+ \frac{1}{2}\sinh{t}\sqrt{1 + \sinh^2{t}} + \frac{3}{8}t\right] + C

     = \frac{x^4}{16}\left[\frac{1}{8}(2\sinh^2{t} + 1)\sinh{t}\sqrt{1 + \sinh^2{t}} + \frac{1}{2}\sinh{t}\sqrt{1 + \sinh^2{t}} + \frac{3}{8}t\right] + C

     = \frac{x^4}{16}\left\{\frac{1}{8}\left[2\left(\frac{y}{x}\right)^2 + 1\right]\left(\frac{y}{x}\right)\sqrt{1 + \left(\frac{y}{x}\right)^2} + \frac{1}{2}\left(\frac{y}{x}\right)\sqrt{1 + \left(\frac{y}{x}\right)^2} + \frac{3}{8}\sinh^{-1}\left(\frac{y}{x}\right)\right\} + C
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