Chapter 15, Problem 4P

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

If **a**, **b**, and **c** are constant vectors, **r** is the position vector *x* **i** + *y* **j** + *z* **k**, and *E* is given by the inequalities 0 ≤ **a** · **r** ≤ *α*, 0 ≤ **b** · **r** ≤ *β*, 0 ≤ **c** · **r** ≤ *γ*, show that

To determine

**To show:**

Explanation

**Given:**

The constant vectors are denoted by **a,b** and **c.**

The position vector **r.**

The region *E* is

**Property used:** Change of Variable

Change of Variable in double integral is given by,

**Calculation:**

Use the transformation *E* can be rewritten as

Obtain the Jacobian by the definition of the dot and cross product.

By using the equation (1) the given integral will be,

Integrate the above integral with respect to *u* and apply the limit.

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