11.6 Cylinders and Quadric Surfaces


1.(a) What does the equation represent as a curve in ?

 Sol) Equation represents a parabola of slope passing through origin in .


 (b) What does it represent as a surface in ?

 Sol) The equation of the graph is , which doesn't involve in . This means that any vertical plane with equation (parallel to the plane) intersects the graph in a curve with , that is, a parabola. Below figure shows how the graph is formed by taking the parabola in the plane and moving it in the direction of the axis. So the graph is a surface, called a parabolic cylinder, made up of infinitely many shifted copies of the same parabola.


 


 (c) What does the equation represent?

 Sol) also represents a parabolic cylinder, this time with axis the axis in .


2.(a) Sketch the graph of as a curve in .

http://matrix.skku.ac.kr/cal-lab/cal-11-6-2.html

 Sol)

(You may do it with Sage in http://math1.skku.ac.kr/ . Open resources in http://math1.skku.ac.kr/pub/)

 

plot(exp(2*x),(x,-1,1))


 


 (b) Sketch the graph of as a surface in .

 Sol)

 

var('x,y,z');

implicit_plot3d(y==exp(2*x),(x,-1,1),(y,-1,1),(z,-1,1))


 


 (c) Describe and sketch the surface .

 Sol)

 

var('x,y,z');

implicit_plot3d(z==exp(2*y),(x,-1,1),(y,-1,1),(z,-1,1))


 

3-5. Describe and sketch the surface.


3. .      

http://matrix.skku.ac.kr/cal-lab/cal-11-6-3.html 

 Sol)

 

var('x, y, z')

implicit_plot3d(y^2+5*z^2-5==0, (x, -3, 3), (y, -3, 3), (z, -3, 3), opacity=0.5)


 


4. .      

http://matrix.skku.ac.kr/cal-lab/cal-11-6-4.html

 Sol)

 

var('x,y')

z=cos(y)

plot3d(z, (x, -2, 2), (y, -pi, pi))


 


5. .

http://matrix.skku.ac.kr/cal-lab/cal-11-6-5.html

 Sol)

 

var('x, y, z')

implicit_plot3d(y*z==3, (x, -3, 3), (y, -10, 10), (z, -10, 10), opacity=0.5)

 


6-10. Find the traces of the given surface in , , . Then, identify the surface and sketch it.


6. .       

http://matrix.skku.ac.kr/cal-lab/cal-11-6-6.html

 Sol)

var('x,y,z');

implicit_plot3d(x^2+9*y^2+9*z^2==9,(x,-3,3),(y,-3,3),(z,-3,3), opacity=0.5)


 


7. .

 Sol) The trace in are ellipses of the form , , the trace in are parabolas of the form , and the trace in are parabolas of the form .

      Combining these traces we form the graph.


 


8. .

http://matrix.skku.ac.kr/cal-lab/cal-11-6-8.html

 Sol)

 

var('x,y,z');

implicit_plot3d(25*x^2+z^2==100+4*y^2,(x,-3,3),(y,-3,3),(z,-3,3), opacity=0.5)


 


9. .

 Sol) The trace in are hyperbolas of the form , the trace in are circles of the form , , and the trace in are hyperbolas of the form .

     Combining these traces we form the graph.


 


10. .

http://matrix.skku.ac.kr/cal-lab/cal-11-6-10.html

 Sol)

 

var('x,y,z');

implicit_plot3d(y^2+4*z^2-x==0,(x,-3,3),(y,-3,3),(z,-3,3), opacity=0.5)

 


11-14. Reduce the equation to one of the standard forms, classify the surface, and sketch it.


11. .

 Sol) Dividing both sides into 15 gives , an elliptic paraboloid with vertex and axis the horizontal line .


 


12. .

http://matrix.skku.ac.kr/cal-lab/cal-11-6-12.html

 Sol)

 

var('x,y,z');

implicit_plot3d(z^2==9*x^2+4*y^2-5,(x,-3,3),(y,-3,3),(z,-3,3), opacity=0.5)


 

13. .    

 Sol) Completing squares in and gives or

      , a hyperboloid of one sheet.


 


14. .

http://matrix.skku.ac.kr/cal-lab/cal-11-6-14.html

 Sol)

 

var('x,y,z');

implicit_plot3d(x^2==3*y^2+2*z^2,(x,-3,3),(y,-3,3),(z,-3,3), opacity=0.5)


 

 

15. .

http://matrix.skku.ac.kr/cal-lab/cal-11-6-15.html

 Sol)

var('x,y,z')

implicit_plot3d((x^2+9/4*y^2+z^2-1)^3-x^2*z^3-9/80 *y^2*z^3==0, (x, -1.5, 1.5), (y, -1.5,1.5), (z,-1.5,1.5), color='red', plot_points=80,smooth=True).show()

 


16.Sketch the region bounded by the surfaces and for .

http://matrix.skku.ac.kr/cal-lab/cal-11-6-16.html

 Sol)

 

var('x, y, z')

z=(x^2+y^2)^(1/2)

plot3d(z, (x, -2, 2), (y, -2, 2), (z, 2, 4))


 


17. Find an equation for the surface obtained by rotating the parabola about the -axis.


18. Find an equation for the surface consisting of all points for which the distance from to the -axis is twice the distance from to the -plane. Identify the surface.

 Sol) Let be an arbitrary point whose distance from th axis is twice its distance from the plane. The distance from to the axis is and the distance from to the plane() is .

      Thus         .

      So, the surface is a right circular cone with vertex the origin and axis the axis.


19. Find an equation for the surface consisting of all points that are equidistant from the point and the plane .