Q. 46 - Calculus

Question

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

Exercise 36 with the order dy dz dx.

Step-by-Step Solution

Answer

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

${\int }_{-3}^2{\int }_4^8{\int }_{-1}^{5}f(x,y,z)dydzdx$

Step 1. Given Information.

We are given,

${\int }_{-1}^5{\int }_{-3}^2{\int }_4^{8}f(x,y,z)dzdxdy$

Step 2. Rewriting the integral.

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

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

The limit of y  alters from $-1 \text{to} 5$

The limit of z alters from $4 \text{to} 8$ 

The limit of x alters from $-3 \text{to} 2$

Changing the order of integration with respect to dy dz dx 

${\int }_{-1}^5{\int }_{-3}^2{\int }_4^{8}f(x,y,z)dzdxdy={\int }_{-3}^2{\int }_4^8{\int }_{-1}^{5}f(x,y,z)dydzdx$