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Math Help - multiple integrals

  1. #1
    Junior Member
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    multiple integrals

    hello,

    I would like to find the volume in the first octant bounded by the
    parabloid z= 1 - x^2 - y^2 and the plane x+y=1

    the limits of the integral w.r.t. y are y = 0 to y=1
    ... .... .... w.r.t. x are x= 1-y to x= (1-y^2)^(1/2)
    ... ... .... w.r.t z are z=0 to z= 1 - x^2 - y^2

    I can do the differentiation w.r.t z and x but it gets to difficult for me to do it w.rt. y.

    can someone help me? much appreciated!
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  2. #2
    Senior Member DeMath's Avatar
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    Quote Originally Posted by pepsi View Post
    hello,

    I would like to find the volume in the first octant bounded by the
    parabloid z= 1 - x^2 - y^2 and the plane x+y=1

    the limits of the integral w.r.t. y are y = 0 to y=1
    ... .... .... w.r.t. x are x= 1-y to x= (1-y^2)^(1/2)
    ... ... .... w.r.t z are z=0 to z= 1 - x^2 - y^2

    I can do the differentiation w.r.t z and x but it gets to difficult for me to do it w.rt. y.

    can someone help me? much appreciated!
    z = 1 - {x^2} - {y^2},{\text{ }}x + y = 1,{\text{ }}x \geqslant 0,{\text{ }}y \geqslant 0.

    D = \left\{ {\left( {x,y} \right):{\text{ }}1 - x \leqslant y \leqslant \sqrt {1 - {x^2}} ,{\text{ }}0 \leqslant x \leqslant 1} \right\}.

    V = \iint\limits_D {\int\limits_0^{1 - {x^2} - {y^2}} {dzdydx} } = \int\limits_0^1 {\int\limits_{1 - x}^{\sqrt {1 - {x^2}} } {\left( {1 - {x^2} - {y^2}} \right)dydx} }  =

    = \int\limits_0^1 {\left. {\left( {y - {x^2}y - \frac{{{y^3}}}<br />
{3}} \right)} \right|_{y = 1 - x}^{y = \sqrt {1 - {x^2}} }dx}  = \int\limits_0^1 {\left( {\frac{2}<br />
{3}\sqrt {{{\left( {1 - {x^2}} \right)}^3}}  - \frac{4}<br />
{3}{x^3} + 2{x^2} - \frac{2}<br />
{3}} \right)dx}  =

    = \frac{2}<br />
{3}\int\limits_0^1 {\sqrt {{{\left( {1 - {x^2}} \right)}^3}} dx}  - \left. {\left( {\frac{{{x^4}}}<br />
{3} - \frac{{2{x^3}}}<br />
{3} + \frac{{2x}}<br />
{3}} \right)} \right|_0^1 = \frac{2}<br />
{3}\int\limits_0^1 {\sqrt {{{\left( {1 - {x^2}} \right)}^3}} dx}  - \frac{1}<br />
{3} =

    = \left\{ \begin{gathered}<br />
  x = \sin t, \hfill \\<br />
  dx = \cos t\,dt, \hfill \\<br />
  0 \leqslant t \leqslant \pi /2 \hfill \\ <br />
\end{gathered}  \right\} = \frac{2}<br />
{3}\int\limits_0^{\pi /2} {{{\cos }^4}t\,dt}  - \frac{1}<br />
{3} = \frac{2}<br />
{3}\int\limits_0^{\pi /2} {{{\left( {\frac{{1 + \cos 2t}}<br />
{2}} \right)}^2}dt}  - \frac{1}<br />
{3} =

    = \frac{1}<br />
{6}\int\limits_0^{\pi /2} {\left( {1 + 2\cos 2t + {{\cos }^2}2t} \right)dt}  - \frac{1}<br />
{3} = \frac{1}<br />
{6}\int\limits_0^{\pi /2} {\left( {1 + 2\cos 2t + \frac{{1 + \cos 4t}}<br />
{2}} \right)dt}  - \frac{1}<br />
{3} =

    = \frac{1}<br />
{{12}}\int\limits_0^{\pi /2} {\left( {3 + 4\cos 2t + \cos 4t} \right)dt}  - \frac{1}<br />
{3} = \frac{1}<br />
{{12}}\left. {\left( {3t + 2\sin 2t + \frac{1}<br />
{4}\sin 4t} \right)} \right|_0^{\pi /2} - \frac{1}<br />
{3} =

    = \frac{1}<br />
{{12}} \cdot \frac{{3\pi }}<br />
{2} - \frac{1}<br />
{3} = \frac{\pi }<br />
{8} - \frac{1}<br />
{3}{\text{ }}\left( {{\text{cubic units}}} \right).
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