VC4.29

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\iiint _{V}Fdv=\iiint _{V}(xi+yj+zk)dxdydz\,

=i\iiint _{V}xdxdydz+j\iiint _{V}ydxdydz+k\iiint _{V}zdxdydz\, =iI_{1}+jI_{2}+kI_{3}\, (say) --(1)

To evaluate above tripple integrals we note that the limits of integration for the region V are z=x^{2}\, to z=4,y=0 to y=6 and x=0 to x=2.

Therefore,I_{1}=\int _{{x=0}}^{{2}}\int _{{y=0}}^{{6}}\int _{{z=x^{2}}}^{{4}}xdxdydz\, =\int _{0}^{2}\int _{0}^{6}x[z]_{{z=x^{2}}}^{{4}}dxdy\, =\int _{0}^{2}\int _{0}^{6}(4x-x^{3})dxdy=\int _{0}^{2}(4x-x^{3})[y]_{0}^{6}dx\, =6\int _{0}^{2}(4x-x^{3})dx=6[2x^{2}-{\frac  {x^{4}}{4}}]_{0}^{2}=6(8-4)=24\, --(2)

I_{2}=\int _{{x=0}}^{{2}}\int _{{y=0}}^{{6}}\int _{{z=x^{2}}}^{{4}}ydxdydz\, =\int _{0}^{2}\int _{0}^{6}y[z]_{{z=x^{2}}}^{{4}}dxdy\, =\int _{0}^{2}\int _{0}^{6}(4-x^{2})ydydz=\int _{0}^{2}(4-x^{2})[{\frac  {y^{2}}{2}}]_{0}^{6}dx\, =18\int _{0}^{2}(4-x^{2})dx=18[4x-{\frac  {x^{3}}{3}}]_{0}^{2}=96\, --(3)

and I_{3}=\int _{{x=0}}^{{2}}\int _{{y=0}}^{{6}}\int _{{z=x^{2}}}^{{4}}zdxdydz\, =\int _{0}^{2}\int _{0}^{6}[{\frac  {z^{2}}{2}}]_{{z=x^{2}}}^{{4}}dxdy\, =\int _{0}^{2}\int _{0}^{6}(16-x^{4})dxdy={\frac  {1}{2}}\int _{0}^{2}[y]_{0}^{6}dx\, =3\int _{0}^{2}(16-x^{4})dx=3[16x-{\frac  {x^{5}}{5}}]_{0}^{2}=3[32-{\frac  {32}{5}}\, ={\frac  {384}{5}}\, --(4)

BY using (2),(3) and (4),(1) becomes,

\iiint _{V}Fdv=24i+96j+{\frac  {384}{5}}k\,

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