Cylinders and Quadratic Surfaces A cylinder is the continuation of a 2-D curve into 3-D.  No longer just a soda can. Ex. Sketch the surface z = x 2.

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Presentation transcript:

Cylinders and Quadratic Surfaces A cylinder is the continuation of a 2-D curve into 3-D.  No longer just a soda can. Ex. Sketch the surface z = x 2.

Ex. Sketch the surface y 2 + z 2 = 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 surface The Graph xy-traceyz-tracexz-trace

Ex Sketch the surface The Graph xy-traceyz-tracexz-trace

Ex Sketch the surface The Graph xy-traceyz-tracexz-trace

Ex Sketch the surface The Graph xy-traceyz-tracexz-trace

Ex Sketch the surface The Graph xy-traceyz-tracexz-trace

Ex Sketch the surface The Graph -trace

Ex Determine the region bounded by and x 2 + y 2 + z 2 = 1 The Graph xy-traceyz-tracexz-trace

Surfaces of rotation x-axis: y 2 + z 2 = [r(x)] 2 y-axis: x 2 + z 2 = [r(y)] 2 z-axis: x 2 + y 2 = [r(z)] 2 Ex. Write the equation of the surface generated by revolving about the y-axis.