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Unformatted text preview: nates. y2 z2 1. SOLUTION Substituting the expressions in Equations 3 into the given equation, we have
2 sin 2 cos 2
2 2 sin 2 sin 2 sin 2 cos 2
2 or 2 cos 2 1 sin 2 sin 2 cos 2 1 cos 2 cos 2 1 EXAMPLE 7 Find a rectangular equation for the surface whose spherical equation is sin sin . SOLUTION From Equations 4 and 3 we have x2 y2 z2 2 sin y ) z2 1
which is the equation of a sphere with center (0, 1 , 0) and radius 1 .
x2 or (y sin
2 EXAMPLE 8 Use a computer to draw a picture of the solid that remains when a hole of radius 3 is drilled through the center of a sphere of radius 4. |||| Most three-dimensional graphing programs
can graph surfaces whose equations are given
in cylindrical or spherical coordinates. As
Example 8 demonstrates, this is often the most
convenient way of drawing a solid. SOLUTION To keep the equations simple, let’s choose the coordinate system so that the
center of the sphere is at the origin and the axis of the cylinder that forms the hole is the
z-axis. We could use either cylindrical or spherical coordinates to describe the solid, but
the description is somewhat simpler if we use cylindrical coordinates. Then the equation of the cylinder is r 3 and the equation of the sphere is x 2 y 2 z 2 16, or
r 2 z 2 16. The points in the solid lie outside the cylinder and inside the sphere, so
they satisfy the inequalities
3 r s16 z 2 To ensure that the computer graphs only the appropriate parts of these surfaces, we ﬁnd
where they intersect by solving the equations r 3 and r s16 z 2 :
z2 s16 3 ? z2 16 9 ? z2 7 ? z s7 The solid lies between z
s7 and z s7, so we ask the computer to graph the surfaces with the following equations and domains:
r z2 2 s7 z s7 0 2 s7 z s7 The resulting picture, shown in Figure 11, is exactly what we want. FIGURE 11 |||| 13.7 s16 0 3 Exercises 1. What are cylindrical coordinates? For what types of surfaces do they provide convenient descriptions?
2. What are spherical coordinates? For what types of surfaces do they provide convenient d...
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This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas at Austin.
- Fall '09