Question

List the six possible orders of integration for the triple integral over the solid region \(Q\) \(\iint_{Q} \int x y z d V\). $$ Q=\left\\{(x, y, z): x^{2}+y^{2} \leq 9,0 \leq z \leq 4\right\\} $$

Short answer

The six possible orders of integration for the given triple integral are: \[dz\,dx\,dy\], \[dz\,dy\,dx\], \[dx\,dy\,dz\], \[dx\,dz\,dy\], \[dy\,dx\,dz\], \[dy\,dz\,dx\].

Step-by-step solution

Identify the integral and region

We are given a triple integral over the solid region \(Q\) of the function \(xyz\), where \(Q=\left\{(x, y, z): x^{2}+y^{2} \leq 9,0 \leq z \leq 4\right\}\). This region is a cylinder.

List the six possible orders of integration

The six possible orders of integration for a triple integral are: \[dz\,dx\,dy\], \[dz\,dy\,dx\], \[dx\,dy\,dz\], \[dx\,dz\,dy\], \[dy\,dx\,dz\], \[dy\,dz\,dx\]. So, one can integrate the variables in the orders listed above.