   Chapter 15.2, Problem 59E

Chapter
Section
Textbook Problem

Use Property 11 to estimate the value of the integral.59. ∬ S 4 − x 2 y 2   d A ,   S   =   { ( x ,     y )   |   x 2   +   y 2   ≤   1 ,   x   ≥   0 }

To determine

To estimate: The value of given double integral by property 11.

Explanation

Property used:

If mf(x,y)M for all (x,y) in D then, mA(D)Df(x,y)dAMA(D) .

Here, the constants are denoted by m, and M.

The area of the given region D is denoted by A(D) .

Given:

The function, f(x,y)=4x2y2 .

The region S is S={(x,y)|x2+y21,x0} .

Calculation:

From the given condition, it is observed that x2+y21 .

Therefore, 0x21,0y21 .

That is, 0x2y21 .

Multiply by 1 , the inequalities becomes,

1x2y20 .

Add 4 to all the terms,

414x2y24+0

34x2y24

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