   Chapter 14.7, Problem 22E

Chapter
Section
Textbook Problem

Mass In Exercises 21 and 22, use cylindrical coordinates to find the mass of the solid Q of density ρ . Q = { ( x , y , z ) :       0 ≤ z ≤ 12 e − ( x 2 + y 2 ) , x 2 + y 2 ≤ 4 , x ≥ 0 ,   y ≥ 0 } ρ ( x , y , z ) = k

To determine

To calculate: The mass of the solid represented by the equation,

Q={(x,y,z):0z12e(x2+y2),x2+y24,x20,y20}

and density provided as ρ(x,y,z)=k

Explanation

Given:

Mass of the solid denoted as Q of density ρ

Q={(x,y,z):0z12e(x2+y2),x2+y24,x20,y20}ρ(x,y,z)=k

Formula Used:

Mass of a solid =ρ(x,y,z)dV

And, volume of solid by triple iterated integration is

V=dVV=rdzdrdθ

Now, rectangular conversion equations of cylindrical coordinates

x=rcosθy=rsinθz=z

Calculation:

Provided, Bounds on z are

0z12e(x2+y2)

By using polar components,

0z12e(r2)

Now, bounds on r are

x2+y24r240r2

By the problem, x and y both should be greater than equal to zero, this condition is applicable only in first quadrant

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