Chapter 15.2, Problem 15E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it’s easier.

**15.**
*D* is bounded by *y* = *x* − 2,.*x* = *y*^{2}

To determine

**To setup:** The iterated integrals for both the orders of integration and then evaluate the double integral by easier method.

Explanation

**Given:**

The function is

The region *D* is bounded by the curves
*.*

**Definition used:**

**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 when *x* varies from 0 to 1*, y* varies from
*x* varies from 1 to 4, *y* varies from *x* âˆ’ 2 to

Similarly, if the region *D* is of type 2, then the region that has to be evaluated is given below in the Figure 2.

Therefore, from Figure 2, it is observed that *x* varies from
*y +* 2and *y* varies from âˆ’1to 2.

So, if the region *D* is of type 1, then the two double integral values has to be evaluated, but if the region *D* is of type 2, then it is enough to evaluate only one double integral value. This is the easier method.

So, the value of double integral is obtained as follows

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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