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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Find the mass of a thin funnel in the shape of a cone z = x 2 + y 2 , 1 ≤ z ≤ 4, if its density function is ρ(x, y, z) = 10 - z.

To determine

To find: The mass of a thin funnel in the shape of a cone.

Explanation

Given data:

ρ(x,y,z)=10z and z=x2+y2 , 1z4 .

Formula used:

m=Sρ(x,y,z)dS (1)

SFdS=DF(rϕ×rθ)dA (2)

rϕ=xϕi+yϕj+zϕk (3)

Substitute 10z for ρ(x,y,z) and x2+y2 for z in equation (1),

m=S(10x2+y2)dS

Modify equation (2).

S(10x2+y2)dS=D(10x2+y2)|rx×ry|dA (4)

Find rx .

Modify equation (3).

rx=xxi+yxj+zxk

Substitute x for x, y for y and x2+y2 for z,

rx=x(x)i+x(y)j+x(x2+y2)k=i+(0)j+(2x2x2+y2)k=i+(xx2+y2)k

Find ry .

Modify equation (3).

ry=xyi+yyj+zyk

Substitute x for x , y for y and x2+y2 for z ,

ry=y(x)i+y(y)j+y(x2+y2)k=(0)i+j+(2y2x2+y2)k=j+(yx2+y2)k

Find rx×ry .

rx×ry=(i+(xx2+y2)k)×(j+(yx2+y2)k)=|ijk10xx2+y201yx2+y2|={(0xx2+y2)i+(0yx2+y2)j+(10)k}=xx2+y2iyx2+y2j+k

Find |rx×ry| .

|rx×ry|=|xx2+y2iyx2+y2j+k|=(xx2+y2)2+(yx2+y2)2+(1)2=x2x2+y2+y2x2+y2+1=x2+y2x2+y2+1

Simplify the equation

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Chapter 16 Solutions

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