Q. 36 - Calculus

Question

Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.

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

Step-by-Step Solution

Answer

The three-dimensional region is,

$\mathrm{ℝ}=\left\{\right(x,y,z)\mid -3\le x\le 2,-1\le y\le 5,4\le z\le 8\}$

Step 1. Given Information

We are given,

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

Step 2. The three dimensional region.

By the definition of triple integral ${\int }_{a_1}^{a_1}{\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}f(x,y,z)dzdydx$ represent the volume of the solid region $\mathrm{ℝ}=\left\{(x,y,z)\mid a_1\le x\le a_2,b_1\le y\le b_2,c_1\le z\le c_2\right\}$

Using this definition,

The given iterated triple integral ${\int }_{-1}^5{\int }_{-3}^2{\int }_4^{8}f(x,y,z)dzdxdy$ represents the volume of the rectangular solid given by $\mathrm{ℝ}=\left\{\right(x,y,z)\mid -3\le x\le 2,-1\le y\le 5,4\le z\le 8\}$

Since by taking the order of the integration in the given iterated integral order that is $dzdxdy$.