anonymous
  • anonymous
triple integral x^2 given vertices
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
\[\int\limits \int\limits \int\limits xyz\]the solid tetrahedron with vertices (0,0,0)(1,0,0)(1,1,0)(1,0,1)
anonymous
  • anonymous
dV
TuringTest
  • TuringTest
what?

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anonymous
  • anonymous
ignore the first part, just the integral
TuringTest
  • TuringTest
you want\[\int\int\int xyzdV\]over the terahedron?
anonymous
  • anonymous
yes
anonymous
  • anonymous
i drew it and i think it's bounded by x=1, y=0 but now I'm stuck
TuringTest
  • TuringTest
let me try to plot the points...
TuringTest
  • TuringTest
|dw:1332796287285:dw|these circles are the points...
TuringTest
  • TuringTest
draw in the shape:|dw:1332796535349:dw|I wrote the equation for each line now we need the general region D under the solid
TuringTest
  • TuringTest
|dw:1332796769210:dw|(that is a separate drawing in of D in the xy-plane in the upper-left) same thing can be done on the z axis, so we have\[0\le y\le x\]\[0\le z\le x\]\[0\le x\le1\]which are our bounds can you take it from here?
anonymous
  • anonymous
do i do it dy dz dx
TuringTest
  • TuringTest
that is one way, yeah
Zarkon
  • Zarkon
I'm not liking those bounds
TuringTest
  • TuringTest
oh no what did I do wrong?
Zarkon
  • Zarkon
using those bounds (1,1,1) is a viable point...but that is not in or on the tetrahedra
TuringTest
  • TuringTest
hm.... so D is wrong
Zarkon
  • Zarkon
I would use \[0\le z\le x-y\]
TuringTest
  • TuringTest
oooooooooooooooh.... right, I get it
anonymous
  • anonymous
how'd you get x-y?
Zarkon
  • Zarkon
I found the equation of the plane on the top part of the tet
Zarkon
  • Zarkon
use 2 vectors...compute cross product to get the normal vector...
Zarkon
  • Zarkon
get the plane -x+y+z=0
anonymous
  • anonymous
ah got cha!
Zarkon
  • Zarkon
can't stay...good luck :)
TuringTest
  • TuringTest
|dw:1332797428957:dw|That's a cool trick zarkon, but I just see it as the fact that I neglected this side of the tetraheda
TuringTest
  • TuringTest
I think....

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