Cylinders and Quadratic Surfaces A cylinder is the continuation of a 2-D curve into 3-D. No longer...
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Transcript of Cylinders and Quadratic Surfaces A cylinder is the continuation of a 2-D curve into 3-D. No longer...
Cylinders and Quadratic SurfacesA cylinder is the continuation of a 2-D curve into 3-D.
No longer just a soda can.
Ex. Sketch the surface z = x2.
Ex. Sketch the surface y2 + z2 = 1.
A quadratic surface has a second-degree equation. The general equation is:
A trace is the intersection of a surface with a plane.
We will use the traces of the quadratic surfaces with the coordinate planes to identify the surface.
Ex Sketch the surface2 2
2 14 9
y zx
The Graph
xy-trace
y
x
yz-trace xz-trace
z
x
z
y
y
z
x
Ex Sketch the surface 2 24z x y
The Graph
xy-trace
y
x
yz-trace xz-trace
z
x
z
y
y
z
x
Ex Sketch the surface
The Graph
2 22 1
4 4
x zy
xy-trace
y
x
yz-trace xz-trace
z
x
z
y
y
z
x
Ex Sketch the surface 2 2z x y
The Graph
xy-trace
y
x
yz-trace xz-trace
z
x
z
y
y
z
x
Ex Sketch the surface 2 2 22 4 0x y z
The Graph
xy-trace
y
x
yz-trace xz-trace
z
x
z
y
y
z
x
Ex Sketch the surface 2 22 6 10 0x z x y
The Graph
-trace
y
x
-trace -trace
z
x
z
y
y
z
x
Ex Determine the region bounded by and x2 + y2 + z2 = 12 2z x y
The Graph
xy-trace
y
x
yz-trace xz-trace
z
x
z
y
y
z
x
Surfaces of rotation
x-axis: y2 + z2 = [r(x)]2
y-axis: x2 + z2 = [r(y)]2
z-axis: x2 + y2 = [r(z)]2
Ex. Write the equation of the surface generated by
revolving about the y-axis.1
yz