Question

In Exercises 45–52, rewrite the indicated integral with the specified order of integration.

Exercise 40 with the order dx dy dz.

Short answer

The given indicated integral with the specified order of integration is,

${\int }_{-4}^0{\int }_0^{1-3x/2}{\int }_{\frac{z}{2}-\frac{2}{3}y-2}^{0}f(x,y,z)dxdydz$

Step-by-step solution

Step 1. Given Information. 

We are given,

${\int }_0^2{\int }_0^{1-3x/2}{\int }_{2x+(4/3)y-4}^{0}f(x,y,z)dzdydx$

Step 2. Rewriting the integral. 

We have to write the given integral in order of dx dy dz .

Take $z=2x+\frac{4}{3}y-4$, which implies $x=\frac{z}{2}-\frac{2y}{3}-2$

The limits become to the following,

z varies from $-4\text{to}0$and

y varies from $0\text{to}3-\frac{3}{2}x$and

x varies from $0\text{to}z-\frac{2}{3}y+2$

Thus, the required integral is,

${\int }_{-4}^0{\int }_0^{1-3x/2}{\int }_{\frac{z}{2}-\frac{2}{3}y-2}^{0}f(x,y,z)dxdydz$