Chapter 15.6, Problem 31E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

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

**31**. *y* =*x*^{2}, *z* = 0, *y* + *2z* = 4

To determine

**To express:** The integral

Explanation

**Given:**

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

**Calculation:**

Let
*E* on *xy, yz* and *zx*-planes.

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

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

From Figure 1, it is observed that *x* varies from
*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 4, *x* varies from
*z* varies from 0 to

Hence,

Therefore,

The variable
*E* on *yz*-plane.

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

From Figure 2, it is observed that *y* varies from 0 to 4 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 2 , *y* varies from 0 to
*x* varies from

Hence,

Therefore,

The variable
*E* on *z x*-plane. So, set

The graph of the above function is shown below in Figure 3.

From Figure 3, it is observed that *x* varies from
*z* varies from 0 to
*y*, Solve the given equations as below

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