# Thread: Triple integral

1. ## Triple integral

How to evaluate the triple integral where is the solid tetrahedon with vertices ?
thankyou!

2. Here's a similar problem although the points in that post aren't showing up but still it's the same: you first have to calculate a normal to the plane you're integrating under, then derive the equation of the plane, then integrate over the appropriate region in the x-y plane.

http://www.mathhelpforum.com/math-he...trahedron.html

3. Maybe we should work this one through since it's been asked a few times. Using the plot below which is a little hard to visualize unless you can rotate it, I'll calculate two vectors: One goes from the point at (0,3,0) to (0,0,2) and the other goes from (8,0,0) to (0,3,0):

$\displaystyle v_1=<0,3,0>-<8,0,0>=<-8,3,0>$

$\displaystyle v_2=<0,0,2>-<0,3,0>=<0,-3,2>$

In order to calculate a normal to the blue surface, I calculate the cross-product of these two vectors which I've indicated as the line eminating from the origin.

$\displaystyle \left|\begin{array}{ccc} i & j & k \\ -8 & 3 & 0 \\ 0 & -3 & 2 \end{array}\right|$

$\displaystyle n=6i+16j+24k$

I can now derive the equation of the blue surface using the equation of the plane passing through the point (0,3,0) and the normal vector:

$\displaystyle n_x(x-p_x)+n_y(y-p_y)+n_z(z-p_z)=0$

$\displaystyle 6(x-0)+16(y-3)+24(z-0)=0$

or: $\displaystyle z=f(x,y)=\frac{-6x-16(y-3)}{24}$

That line in the back is $\displaystyle y=-3/8x+3$ right?

So then the integral is:

$\displaystyle V=\int_0^8\int_0^{-3/8x+3}\int_0^{f(x,y)} xydzdydx$

Pretty sure anyway.