Q. 35 - Calculus

Question

Describe the three-dimensional region expressed in each iterated integral:

${\int }_{-2}^4 {\int }_2^6 {\int }_0^5 f(x,y,z)dydxdz$

Step-by-Step Solution

Answer

$\mathrm{ℝ}=\left\{\right(x,y,z)\mid 2\le x\le 6,0\le y\le 5,-2\le z\le 4\}$

Step 1. Given information.

We have been given a three-dimensional region: 

${\int }_{-2}^4 {\int }_2^6 {\int }_0^5 f(x,y,z)dydxdz$

We have to describe it in the iterated integral.

Step 2. Evaluate.

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 $\mathrm{ℝ}=\left\{(x,y,z)\mid a_1\le x\le a_2,b_1\le y\le b_2,c_1\le z\le c_2\right\}$

The given triple integral represents the volume of the rectangular solid given by $\mathrm{ℝ}=\left\{\right(x,y,z)\mid 2\le x\le 6,0\le y\le 5,-2\le z\le 4\}$