Consider the solid in xyz-space, which contains all points (x, y, z) whose z-coordinate satisfies...

Marenonigt

Marenonigt

Answered

2021-12-18

Consider the solid in xyz-space, which contains all points (x, y, z) whose z-coordinate satisfies
0z4x2y2
Which statements do hold?
a) The solid is a sphere
b) The solid is apyramid
c) Its volume is 8π
d) Its volume is 16π3

Answer & Explanation

Carl Swisher

Carl Swisher

Expert

2021-12-19Added 28 answers

Step 1
dv
dzdydx
z=0 to z=4x2y2
(04x2y2dz)dydx
(4x2y2)dydx
Change in folar co-ordinates, tre get
x=rcosθ
y=rsinθ
r=0 to r=2
dxdy=rdrdθ
θ=0 to θ=2π
02π02(4r2)rdrdθ
Let 4r2=trdr=dt2
t=4 to t=44=0
02π(40tdt2)dθ1202π(04tdt)dθ
1202π(t22)04dθ1202π8dθ
402πdθ4(2π)
2π

Carl Swisher

Carl Swisher

Expert

2021-12-20Added 28 answers

Here, take limit: 04x2y2
x2+y24
And we know,
(4x2y2)max=4 when x=0 and y=0
So, 0z4
For each value of z0, it will be a disk of at each value of zR.
Now, volume in =πBr2h
=πB×(2)2×(4)=16πB

nick1337

nick1337

Expert

2021-12-28Added 573 answers

Step 1
Consider the solid  in xyz-space which contains all points (x,y,z) whose z-coordinate satisfies 0z4x2y2
Step 2
The solid is paraboloid.
We use cylindrical coordinates

x=rcost,y=rsint,0z4r2,0t2π
and z=0 gives 4r2=0
r2=4
r=2
0r2
The volume of the solid is given by
V=SdV=0202π04r2rdzdtdr=0202π|z|04r2rdtdr=0202π(4r2)rdtdr=02r(4r2)[t]02πdr=02r(4r2)[2π0]dr=2π02(4rr3)dr=2π[2r2r44]02=2π[2×222440]=2π[84]=8π
Thus, volume of the solid is 8π.
Hence, option (c) is correct.

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