Question

List the six possible orders of integration for the triple integral over the solid region \(Q\) \(\iint_{Q} \int x y z d V\). $$ Q=\\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq x, 0 \leq z \leq 3\\} $$

Short answer

The six possible orders of integration are: \(\{dxdydz, dxdzdy, dydxdz, dydzdx, dzdxdy, dzdydx\}\), but for the given conditions, the only correct permutations of integration are: \(\{dzdydx, dydxdz\}\).

Step-by-step solution

Understand the Limits

Firstly, identify how the variables are bounded. Looking at the given conditions, \(0 \leq x \leq 1,0 \leq y \leq x, 0 \leq z \leq 3\). These conditions tell us that \(x\) is simply bounded by \(0\) and \(1\), \(y\) is bounded by \(0\) and \(x\), indicating that it relies on the value of \(x\), and \(z\) is bounded by \(0\) and \(3\).

Create Orders of Integration

Understanding the limits, one can form six possible permutations of \(x\), \(y\), and \(z\) for integration. These permutations are: \(\{dxdydz, dxdzdy, dydxdz, dydzdx, dzdxdy, dzdydx\}\).

Verify Orders of Integration

However, not all orders may respect the bounds described. So, the proper bounds for each order of integration should be analyzed. If they respect the original conditions then they qualify.

Determine Correct Orders of Integration

In this composition of bounds, it's noted that the only valid permutations of orders of integration are: \(\{dzdydx, dydxdz\}\), where \(dzdydx\) implies \(0\leq x\leq 1\), \(0 \leq y \leq x\), and \(0 \leq z \leq 3\), and \(dydxdz\) implies \(0\leq z\leq 3\), \(0\leq x\leq 1\), and \(0 \leq y \leq x\).