triple integral pyramid bounds

Question

I am still confused about how to set up bounds for double and triple integrals. My task is to set up bounds for a function that is a pyramid with edge coordinates $(5,+-5, 0)$, $(-5,+-5,0)$, $(0,0,4)$. it follows the order of $dydxdz$. I have already set the bound for $dz$ . it is $0$ to $4$. Can anyone show me how to solve it and explain how I need to think when setting up bounds. Thanks

Answer 1

You have to express both x and y as functions of z, since z is the last bounds for the integral. As z goes from 0 to 4, both x and y goes from -5 to 5. So the functions will be:

$$x = \frac{5}{4}z$$ $$y = \frac{5}{4}z$$

and then the integral may be set up as:

$$\int_0^4 \int_{-\frac{5}{4}z}^{\frac{5}{4}z} \int_{-\frac{5}{4}z}^{\frac{5}{4}z} 1 dy dx dz$$

But, the symmetry argument would be easier... there is a easy formula for the volume of a pyramid.