A bowl is described by by the inequalities z &#x2265;<!-- ≥ --> x 2 </msup>

prensistath

prensistath

Answered question

2022-05-28

A bowl is described by by the inequalities
z x 2 + y 2 1 , z x 2 + y 2 + 1 , z 0.
Sketch the bowl and compute its volume.

Answer & Explanation

1c2ru3x

1c2ru3x

Beginner2022-05-29Added 9 answers

Initially, it is always practical to draw a map of the area by hand x  y, y  z and x  z planes.
Note that for example that for y = 0 that is x  z (and similarly for x = 0) plane the first region is
z  x 2  1 that area is located above the parabola.
z = x 2  1
and in x-y plane
x 2 + y 2  1 that area is located inside the circle.
x 2 + y 2 = 1
and so on.
Regarding the second for y = 0 (and similarly for x = 0)
z  x 2 + 1 = | x | + 1 ,  z  0 that is the region under the cone z = | x | + 1 z = | x | + 1 and above x axis
and for z = k > 1
k  x 2 + y 2 + 1    x 2 + y 2  ( k  1 ) 2 is the area beyond the circle x 2 + y 2 = ( k  1 ) 2 

Kash Brennan

Kash Brennan

Beginner2022-05-30Added 4 answers

The first region's surface's boundary is x 2 + y 2 = 1 + z. For z = K >  1 the section of such surface is a circle of radius r = x 2 + y 2 = 1 + K , the vertex at the center of the first region's paraboloid with circular section at z =  1.
The second region is limited by the surface x 2 + y 2 = z  1 that, for z = H > 1 is a circle of radius r = x 2 + y 2 = H  1. Hence, this area is a cone with a circular cross-section and a vertex at z = 1.
Given the symmetry around the z axis, by adopting cylindrical coordinates, the issue becomes straightforward.

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