# Thread: Volume of a pyramid

1. ## Volume of a pyramid

I have two different problems that are built around the volume of the following:

A three sided pyramid has sides of the following planes y= 0, y- x = 4, 2x + y + z = 4
The base sits at the plane z = -6

I have tried to draw a sketch but am still having trouble with this. I have the base being a platform at z = -6 and one side is along the y axis ( y = 0)

I believe my function should be 4 - 2x - y and that I am dealing with a double integral.

Would someone be willing to sketch this? Or help me with what my limits for x and y are?

2. Originally Posted by Frostking
I have two different problems that are built around the volume of the following:

A three sided pyramid has sides of the following planes y= 0, y- x = 4, 2x + y + z = 4
The base sits at the plane z = -6

I have tried to draw a sketch but am still having trouble with this. I have the base being a platform at z = -6 and one side is along the y axis ( y = 0)

I believe my function should be 4 - 2x - y and that I am dealing with a double integral.

Would someone be willing to sketch this? Or help me with what my limits for x and y are?
Here's the sketch:

Now, to find the limits of integration:

If your dealing with a triple integral, we see that $-6\leqslant z\leqslant 4-2x-y$. Now, when $y=0$, $y-x=4\implies -x=4\implies x=-4$ and when $y=0,z=-6$, $2x+y+z=4\implies 2x-6=4\implies 2x=10\implies x=5$. Thus, $-4\leqslant x\leqslant 5$. Finally, we can conclude that $0\leqslant y\leqslant x+4$

So your triple integral would be $\int_{-4}^5\int_0^{x+4}\int_{-6}^{4-2x-y}\,dz\,dy\,dx$

The double integral would be $\int_{-4}^5\int_0^{x+4}\left(10-2x-y\right)\,dy\,dx$, which is a result from the triple integral.

Does this make sense?

3. ## Limits of integration

Your sketch and detailed explanation helped a lot. I am still struggling with how to envision these shapes. Geometry and spacial problems are tough for me. Thanks very much for your time and effort. I will study your post and try to come up with the limits on my own. Frostking