Chapter 15.6, Problem 32E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Express the integral
*E* is the solid bounded by the given surfaces.

**32**. *x* = 2, *y = 2, z* = 0, *x* + *y –* 2*z* = 2

To determine

**To express:** The integral

Explanation

**Given:**

The region *E* is the solid bounded by the surfaces,

**Calculation:**

Let *E* be the solid and if *E* on *xy*, *yz*, and *zx*-planes.

The variable *E* on *xy*-plane. So, set

Then, the equation becomes,

The graph of the above function is shown below in Figure 1.

From Figure 1, it is observed that *x* varies from 0 to 2 and *y* varies from *z*, solve the given equations as below.

Hence,

Therefore,

Also, from Figure 1, it is observed that *y* varies from 0 to 2, *x* varies from *z* varies from 0 to

Hence,

Therefore,

The variable *E* on *yz*-plane. So, set

The graph of the above function is shown below in Figure 2.

From Figure 2, it is observed that *y* varies from 0 to 2 and *z* varies from 0 to *x*, solve the given equations as below.

Hence,

Therefore,

Also, from Figure 2, it is observed that *z* varies from 0 to 1, *y* varies from *x* varies from

Hence,

Therefore,

The variable *E* on *zx*-plane

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