> read(`c:/othermaple/plot3d_region02.txt`);

Warning, the name changecoords has been redefined

Warning, the name arrow has been redefined

> help_plot3d_region();

EXAMPLE: > gr0:=plot3d_region(x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2),z=-1+sqrt(x^2+y^2)..sqrt(1-x^2-y^2));

         > display(gr0); display([gr0[1],gr0[2]); display([op(3..6,gr0)]);

         to describe an ice cream cone.

	 The fourth argument (20 in the example) controls coarseness of the picture.

	 Time and memory requirements can be proportional to the cube of this number.

	 A list of 6 plot structures is returned.

For all these, one would want to sketch the bounding surfaces and regions before

trying to set up the limits of integration. (The function plot3d_region is really

visualizing the limits supplied, not figuring out the right limits.)

However, to give you a better feel for what we are seeking in setting up a triple

integral, spend a little time thinking at how the solid pictured arises from

the problem specification and the limits supplied.

Remember that in the triple integral ,

1. For fixed values of x and y, the limits g(x,y) and h(x,y) describe the

intersection with the solid of the corresponding line parallel to the z axis.

2. The outer two limits of integration (after the inner z integral is done)

describe the projection of the solid into the xy plane.

Cube. (The Simplest Case.)

Unit Cube 0 <=x <=1,0 <=y<=1,0 <=z <=1.

> gra:=plot3d_region(x=-1..1,y=-1..1,z=-1..1,11):
display(gra); display([gra[1],gra[2]],title=`Top and Bottom`); display([op(3..6,gra)],title=`Sides`);

Solid with Six Pieces to its Boundary. (The Generic Elementary Region.)

Region in the first octant bounded above by z=x^2+y^2+1, below by z=0,

and on the sides by the xz plane, yz plane, the plane y=2 as well as the

parabolic cylinder x=(y-1)^2+1.

> gr0:=plot3d_region(y=0..2,x=0..(y-1)^2+1,z=0..1+x^2+y^2,20):
display(gr0); display([gr0[1],gr0[2]],title=`Top and Bottom`); display([op(3..6,gr0)],title=`Sides`);

Tetrahedron.

Solid Bounded by the planes x=0, y=0, z=0, 2x+y+3z=6.

> gr5:=plot3d_region(x=0..3,y=0..6-2*x,z=0..(6-2*x-y)/3,20):
display(gr5); display([gr5[1],gr5[2]],title=`Top and Bottom`); display([op(3..6,gr5)],title=`Sides`);

Between Two Paraboloids.

Solid Bounded by the paraboloids z=x^2+2y^2 and z=12-2*x^2-y^2.

> gr6:=plot3d_region(x=-2..2,y=-sqrt(4-x^2)..sqrt(4-x^2),z=x^2+2*y^2..12-2*x^2-y^2,20):
display(gr6); display([gr6[1],gr6[2]],title=`Top and Bottom`); display([op(3..6,gr6)],title=`Sides`);

>

Cylinder Cut By Some Planes.

Solid in the first octant cut off by the plane z=y and the cylinder 1=x^2+y^2.

> gr2:=plot3d_region(x=0..1,y=0..sqrt(1-x^2),z=0..y,20):
display(gr2); display([gr2[1],gr2[2]],title=`Top and Bottom`); display([op(3..6,gr2)],title=`Sides`);

Solid Bounded by Some Planes.

Solid Bounded by the planes x=0, y=0, z=0, y+z=1, x+z=1.

(Looking first at the graphs helps here.)

> g2:=plot3d(1-x,x=0..2,y=0..2,color=red):
g3:=plot3d(1-y,x=0..2,y=0..2,color=blue):
display([g2,g3],axes=normal);

Since the top of the solid is sometimes z=1-x and sometimes z=1-y, it is natural to use two pieces.

> gr3:=plot3d_region(x=0..1,y=0..x,z=0..1-x,20):
gr4:=plot3d_region(x=0..1,y=x..1,z=0..1-y,20):

display([op(gr3),op(gr4)],title=`The Two Elementary Regions Combined`);
display(gr3); display([gr3[1],gr3[2]],title=`Top and Bottom`); display([op(3..6,gr3)],title=`Sides`);

gr4:=plot3d_region(x=0..1,y=x..1,z=0..1-y,20):
display(gr4); display([gr4[1],gr4[2]],title=`Top and Bottom`); display([op(3..6,gr4)],title=`Sides`);

Between Two Parabolic Cylinders

The Solid Bounded Above by z=1-x^2 and Below by z=y^2.

(Looking first at the graphs helps here.)

> g0:=plot3d(1-x^2,x=-2..2,y=-2..2,color=red):
g1:=plot3d(y^2,x=-2..2,y=-2..2,color=blue):
display([g0,g1],axes=normal);

> gr1:=plot3d_region(x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2),z=y^2..1-x^2,20):
display(gr1); display([gr1[1],gr1[2]],title=`Top and Bottom`); display([op(3..6,gr1)],title=`Sides`);

>