CALC 3: eval. triple int. xy dV w/ E a solid tetrahedron

[think.ms.tink]

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)

 

[Deleted member 4993]

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.

 

[galactus]

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.