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Q. 35

Expert-verifiedFound in: Page 1055

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

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

${\int}_{-2}^{4}\u200a{\int}_{2}^{6}\u200a{\int}_{0}^{5}\u200af(x,y,z)dydxdz$

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

We have been given a three-dimensional region:

${\int}_{-2}^{4}\u200a{\int}_{2}^{6}\u200a{\int}_{0}^{5}\u200af(x,y,z)dydxdz$

We have to describe it in the iterated integral.

By the definition of triple integral ${\int}_{{a}_{1}}^{{a}_{1}}\u200a{\int}_{{b}_{1}}^{{b}_{2}}\u200a{\int}_{{c}_{1}}^{{c}_{2}}\u200af(x,y,z)dzdydx$ represent the volume of the solid region $\mathrm{\mathbb{R}}=\left\{(x,y,z)\mid {a}_{1}\le x\le {a}_{2},{b}_{1}\le y\le {b}_{2},{c}_{1}\le z\le {c}_{2}\right\}$

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

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