Q. 49 - Calculus

Question

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

Exercise 39 with the order dz dy dx.

Step-by-Step Solution

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 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 $\mathrm{dzdydx}$,

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