Question

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

Exercise 38 with the order dx dy dz.

Short answer

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

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

Step-by-step solution

Step 1. Given Information. 

We are given,

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

Step 2. Rewriting the integral. 

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

So, the limit of the integration will vary as follows:

The limit of xalters from $0\text{to}3$

The limit of yalters from $0\text{to}2$

The limit of zalters from $0\text{to}\frac{4x}{3}$

When $z=\frac{4x}{3}$,

$x=\frac{3z}{4}$

Combining the above limits, x varies from $0$to $\frac{3z}{4}$.

Step 3. Rewriting the integral. 

The limits of y varies from $0$to $2$. There will be no change.

Calculate the limits of z.

When $x=0$,

$$\begin{gathered} z=\frac{4x}{3} \\ =\frac{{\displaystyle 4\left(0\right)}}{{\displaystyle 3}} \\ =0 \end{gathered}$$

When $x=3$,

$$\begin{gathered} z=\frac{4x}{3} \\ =\frac{{\displaystyle 4\left(3\right)}}{{\displaystyle 3}} \\ =4 \end{gathered}$$

Hence the indicated integral with the specified order of integration will be,

${\int }_0^2{\int }_0^3{\int }_{4x/3}^{4}f(x,y,z)dzdxdy={\int }_0^4{\int }_0^2{\int }_0^{3z/4}f(x,y,z)dxdydz\text{.}$