   Chapter 14.7, Problem 21E

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 ≤ 9 − x − 2 y ,   x 2 + y 2 ≤ 4 } ρ ( x , y , z ) = k x 2 + y 2

To determine

To calculate: The provided mass of the solid

Q={(x,y,z):0z9x2y,x2+y24}

Whose density is provided as, ρ(x,y,z)=kx2+y2

Explanation

Given:

The provided mass of the solid Q of density ρ such that

And, Q={(x,y,z):0z9x2y,x2+y24}ρ(x,y,z)=kx2+y2

Formula Used:

Volume of solid by triple iterated integration is

V=dVV=rdzdrdθ

And, rectangular conversion equations of cylindrical coordinates

x=rcosθy=rsinθz=z

And, Mass of a solid =ρ(x,y,z)dV

Calculation:

Provided, Bounds on z are

0z9x2y

By using polar components,

0z9rcosθ2rsinθ

Now, bounds on r are

x2+y24r240r2

Again, bounds on θ are

0θ2π

Therefore, Mass (m) of the solid is:

m=ρ(x,y,z)dV=02π

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