Chapter 15.2, Problem 13E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Express *D* as a region of type I and also as a region of type II. Then evaluate the double integral in two ways.

**13.**
*D* is enclosed by the lines *y* = *x, y* = 0, *x* = 1

To determine

**To express:** The given region *D* as of type 1 and type 2 and then evaluate the double integral*.*

Explanation

**Given:**

The function is

The region *D* is enclosed by the lines *y = x*, *y =* 0 and *x* = 1.

**Definition used:**

**Region of type 1:**

**Region of type 1:**

A plane region *D* is said to be of type 1 if it lies between two continuous functions of *x*.

That is,
*x*.

**Region of type 2:**

A plane region *D* is said to be of type 2 if it lies between two continuous functions of *y*.

That is,
*y*.

If the region *D* is of type 1, then the region that has to be evaluated is given below in the Figure 1.

Therefore, from Figure 1, it is observed that *x* varies from 0 to 1 and *y* varies from 0 to *x*.

So, the value of the double integral is computed as follows.

Integrate with respect to *x*,

Thus, the value of the double integration when considering *D* as a region in type 1 and type 2 is

Sect-15.1 P-1ESect-15.1 P-2ESect-15.1 P-3ESect-15.1 P-4ESect-15.1 P-5ESect-15.1 P-6ESect-15.1 P-7ESect-15.1 P-8ESect-15.1 P-9ESect-15.1 P-10E

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