Q. 50 - Calculus

Question

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

Exercise 40 with the order dx dy dz.

Step-by-Step Solution

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 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,

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