Chapter 15, Problem 33RE

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Calculate the value of the multiple integral.

**33**.
*E* lies above the plane *z =* 0, below the plane *z* = *y,* and inside the cylinder *x*^{2} + *y*^{2} = 4

To determine

**To calculate:** The given triple integral.

Explanation

**Given:**

The function is

The region *E* lies above the plane

**Formula used:**

If
*E* given by

If
*z*, then,

**Calculation:**

Since the equation of the cylinder is given, solving with cylindrical coordinates is a wise choice. From the given conditions, *r* varies from 0 to 2,
*z* varies from 0 to *y* and
*z* varies from 0 to

First compute the integral with respect to *z* and apply the limit.

Use the equation (2) to separate the integral

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