Question

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

Exercise 39 with the order dz dy dx.

Short answer

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

${\int }_0^2{\int }_0^{3-\frac{3}{2}x}{\int }_0^{1-\frac{1}{2}x-\frac{1}{3}y}f(x,y,z)dzdydx$

Step-by-step solution

Step 1. Given Information. 

We are given,

${\int }_0^4{\int }_0^2{\int }_0^{3z/4}f(x,y,z)dxdydz$

Step 2. Rewriting the integral. 

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

From the given integral the x varies from $x=0\text{to}x=2-\frac{2}{3}y-2z$and

y varies from $y=0\text{to}y=3$and

z varies from $z=0\text{to}z=1-\frac{y}{3}$

Changing x in terms of y and $z=0\text{to}z=1-\frac{1}{2}y-\frac{1}{3}x$,

Similarly on X-Axis y and z are zero so $x=0\text{to}x=2$,

Changing y in terms of x, $z=0\text{to}z=1-\frac{y}{3}$

Changing the order of integration with respect to $\mathrm{dzdydx}$,

${\int }_0^2{\int }_0^{3-\frac{3}{2}x}{\int }_0^{1-\frac{1}{2}x-\frac{1}{3}y}f(x,y,z)dzdydx$