think.ms.tink
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Evaluate the triple integral xy dV where E is the solid tetrahedon with vertices (0,0,0), (9,0,0), (0,8,0), (0,0,6)
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CALC 3: eval. triple int. xy dV w/ E a solid tetrahedron
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think.ms.tink said:Evaluate the triple integral xy dV where E is the solid tetrahedon with vertices (0,0,0), (9,0,0), (0,8,0), (0,0,6)
First find the limits of your integration.
Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.
You could start by finding the equation of the plane which passes through the points (9,0,0), (0,8,0), (0,0,6)
The equation of this plane is \(\displaystyle 48x+54y+72z=432\).
Solved for z is \(\displaystyle z=\frac{-2x}{3}-\frac{3y}{4}+6\)
Therefore, the z limits of integration are 0 to \(\displaystyle z=\frac{-2x}{3}-\frac{3y}{4}+6\)
To find the y limits, set z=0 in the original plane equation and solve for y.
We get \(\displaystyle y=8-\frac{8x}{9}\)
The x limits are 0 to 9. They are given.
So, we have:
\(\displaystyle \int_{0}^{9} \;\ \int_{0}^{8-\frac{8x}{9}} \;\ \int_{0}^{\frac{-2x}{3}-\frac{3y}{4}+6}dzdydx\)
Once you have this result, compare it to the volume of a tetrahedron formula, \(\displaystyle V=\frac{1}{3}A_{0}h\)
Where \(\displaystyle A_{0}\) is the area of the base and h is the height. See if you get the same. You should.
The equation of this plane is \(\displaystyle 48x+54y+72z=432\).
Solved for z is \(\displaystyle z=\frac{-2x}{3}-\frac{3y}{4}+6\)
Therefore, the z limits of integration are 0 to \(\displaystyle z=\frac{-2x}{3}-\frac{3y}{4}+6\)
To find the y limits, set z=0 in the original plane equation and solve for y.
We get \(\displaystyle y=8-\frac{8x}{9}\)
The x limits are 0 to 9. They are given.
So, we have:
\(\displaystyle \int_{0}^{9} \;\ \int_{0}^{8-\frac{8x}{9}} \;\ \int_{0}^{\frac{-2x}{3}-\frac{3y}{4}+6}dzdydx\)
Once you have this result, compare it to the volume of a tetrahedron formula, \(\displaystyle V=\frac{1}{3}A_{0}h\)
Where \(\displaystyle A_{0}\) is the area of the base and h is the height. See if you get the same. You should.