Triple Integrals and Spherical Coordinates

Copyright © 1999, 2001 by James F. Hurley, University of Connecticut Department of Mathematics, Unit 3009, Storrs CT 06269-3009. All rights reserved.

1. Using the basic sphereplot command . Rcall that Maple's command sphereplot produces immediate plots of surfaces with equations of the form = f ( , ) . The syntax is

sphereplot( f( , ), = .. , = .. ) ;

This basic command is designed to plot spherical-coordinate equations that are solvable for the radial variable Ñ in this case, Ñ and are swept out as the angles and vary over a basic spherical-coordinate region ("spherical-coordinate rectangle") [ , ] ´ [ , ] in the -plane. It thus can plot only one of the three basic equations coordinate variable = constant , namely the sphere = k.

Just as for the basic cylinderplot command, the basic version of sphereplot is powerful enough to give useful plots of many simple regions that arise in spherical-coordinate integration. The following example considers a triple integral that, in Cartesian coordinates, would be quite formidable.
Example 1 . Evaluate , where E is the region between the two spheres and .

Solution. The region is easy to visualize, and to plot if we restrict the angle t o the range [0, ¹], so that removal of the (hidden) sphere of smaller radius does not eliminate it from view. The following routine is just a slight modification of the last one.

> with (plots):
surf1 := sphereplot (2, theta = 0..Pi, phi = 0..Pi, axes = boxed, scaling = constrained):
surf2 := sphereplot (3, theta = 0..Pi, phi = 0..Pi, axes = boxed, scaling = constrained):
xaxis := spacecurve([t, 0, 0, t = -2..3.5, color = magenta]) :
yaxis := spacecurve([0, t, 0, t = -2..3.5, color = magenta]) :
zaxis := spacecurve([0, 0, t, t = -2..3.5, color = magenta]) :
labx := textplot3d([3.6, 0, -.2, `x`], color = magenta):
laby := textplot3d([0,3.6, -.2, `y`], color = magenta):
labz := textplot3d([0, 0, 3.6, `z`], color = magenta):
display(surf1, surf2, xaxis, yaxis, zaxis, labx, laby,labz);

An arrow (like the x -axis) shot radially outward from the origin enters the region E by passing through the sphere of radius 2, and then exits E through the sphere of radius 3. Thus,

=

=

= ,

where the region D is the "spherical-coordinate -rectangle" [0, 2¹] ´ [0, ¹]. Hence, the value of the triple integral is

= = = 10¹.

As before, Maple can check the evaluation, via the TripleInt command in its student package.

> with (student):
value(Tripleint(rho*sin(phi), rho = 2..3, phi = 0..Pi, theta = 0..2*Pi) );

2. More general regions . Recall that the sphereplot command Ñ in a slightly modified form Ñ can plot the graphs of surfaces whose equations are not of the basic form = f ( , ) . The syntax of the extended sphereplot command requires specification of the formulas for all three variables , and . The syntax for the extended sphereplot command is

sphereplot( [ , , ], u = .. , v = .. ) ;

As with the extended cylinderplot command, in this expression the generic symbols u and v stand for the two independent parameters among , , and that vary to define the surface in question. The next example illustrates the usefulness of the extended sphereplot command by revisiting the first example of the Cylindrical Triple Integral handout.

Example 2 . Find the volume of the region E above the graph of z = and below the graph of = 4.

Solution . The first step is to plot the region E between the cone and the sphere, which of course has the simple spherical-coordinate equation = 2. What is the equation of the cone in spherical coordinates? Recall that

z = cos

and = r (of cylindrical Ñ or polar Ñ coordinates). In deriving the formulas for changing coordinates from spherical to rectangular coordinates, a key relation is

r = sin .

The equation z = r of the cone thus transforms to

cos = sin Þ 1 = tan Þ = .

The following routine generates a plot of the region E without the need Ñ which arose in plotting E via cylindrical coordiantes Ñ to determine algebraically the curve of intersection of the sphere = 2 and the cone = . Execute the routine, and then experiment by rotating it to produce a good view.

> with (plots):
surf1 := sphereplot (2, theta = 0..Pi, phi = 0..Pi/2, axes = boxed):
surf2 := sphereplot ([rho, theta, Pi/4], rho = 0..2, theta = 0..Pi):
xaxis := spacecurve([t, 0, 0, t = -2..2.5, color = magenta]) :
yaxis := spacecurve([0, t, 0, t = -2..2.5, color = magenta]) :
zaxis := spacecurve([0, 0, t, t = -1..2.5, color = magenta]) :
labx := textplot3d([2.6, 0, -.2, `x`], color = magenta):
laby := textplot3d([0,2.6, -.2, `y`], color = magenta):
labz := textplot3d([0, 0, 2.6, `z`], color = magenta):
display(surf1, surf2, xaxis, yaxis, zaxis, labx, laby,labz, scaling = constrained);

An arrow shot radially outward from the origin through E emerges from the sphere = 2 . Bearing in mind that dV = d d d , the volume is then

= =

= = .

Maple's TripleInt command from the student package agrees with this calculation:

> with (student):
value(Tripleint(rho^2*sin(phi), rho = 0..2, phi = 0..Pi/4, theta = 0..2*Pi) );