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Expert-verified Found in: Page 1055 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Describe the three-dimensional region expressed in each iterated integral:${\int }_{-2}^{4} {\int }_{2}^{6} {\int }_{0}^{5} f\left(x,y,z\right)dydxdz$

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

See the step by step solution

## Step 1. Given information.

We have been given a three-dimensional region:

${\int }_{-2}^{4} {\int }_{2}^{6} {\int }_{0}^{5} f\left(x,y,z\right)dydxdz$

We have to describe it in the iterated integral.

## Step 2. Evaluate.

By the definition of triple integral ${\int }_{{a}_{1}}^{{a}_{1}} {\int }_{{b}_{1}}^{{b}_{2}} {\int }_{{c}_{1}}^{{c}_{2}} f\left(x,y,z\right)dzdydx$ represent the volume of the solid region $\mathrm{ℝ}=\left\{\left(x,y,z\right)\mid {a}_{1}\le x\le {a}_{2},{b}_{1}\le y\le {b}_{2},{c}_{1}\le z\le {c}_{2}\right\}$

The given triple integral represents the volume of the rectangular solid given by $\mathrm{ℝ}=\left\{\left(x,y,z\right)\mid 2\le x\le 6,0\le y\le 5,-2\le z\le 4\right\}$ ### Want to see more solutions like these? 