Triple Integrals and Cylindrical Coordinates

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

1. Basic surfaces and basic cylidindrical-coordinate regions . Recall Maple's command for plotting surfaces described by a cylindrical-coordinate equation of the form r = f ( theta , z ):

cylinderplot( f( theta , z), theta = alpha .. beta , z = c..d ) ;


It is effective for surfaces lying above a region of the
xy -plane whose polar-coordinate description is a basic "polar rectangle" [ alpha , beta ] [ c , d ]. This is enough for some types of triple-integration problems.

Example 1 . Find the volume (for example, of ice cream) above the graph of z = sqrt(x^2+y^2) and below the sphere with equation x^2+y^2+z^2 = 4.

Solution . The first step is to plot the region E between the cone and the sphere. The following routine does that. In coding the routine, the first step was determining algebraically the curve C of intersection of the cone and the sphere. C is also the boundary of the subregion D of integration in the xy -plane. To get the equation of C , substitute z^2 = x^2+y^2 from the first equation into the second:

x^2+y^2+x^2+y^2 = 4 x^2+y^2 = 2 ,


which is the circle of radius
sqrt(2) centered at the origin. In polar coordinates that is r = sqrt(2) . Note that the equation of the cone transforms to the cylindrical-coordinate equation z = r , and the equation of the sphere transforms to r^2+z^2 = 4. Since all the ice cream lies above the cone in particular, above the xy- plane in solving the equation of the sphere for r only the positive square root is necessary: r = sqrt(4-z^2) . Letting theta range just from 0 to reveals the interior the region, from which it easy to determine the proper limits of integration. Rotation of the initial plot gives a good image of the region E . Execute the routine, and then experiment with rotating it until you have a good view of the region.

> with (plots):
surf1 := cylinderplot (z, theta = 0..Pi, z = 0..sqrt(2), axes = boxed):
surf2 := cylinderplot (sqrt(4 - z^2), theta = 0..Pi, z = 0..2):
xaxis := spacecurve([t, 0, 0, t = -2..2, color = magenta]) :
yaxis := spacecurve([0, t, 0, t = -2..2, color = magenta]) :
zaxis := spacecurve([0, 0, t, t = -1..3, color = magenta]) :
labx := textplot3d([2.1, 0, -.2, `x`], color = magenta):
laby := textplot3d([0,2.1, -.2, `y`], color = magenta):
labz := textplot3d([0, 0, 3.1, `z`], color = magenta):
display(surf1, surf2, xaxis, yaxis, zaxis, labx, laby,labz);

[Maple Plot]


An arrow shot upward through
E enters the region by piercing the cone z = r and exits by emerging from the top of the sphere z = sqrt(4-r^2) . Bearing in mind that dV = r dr d theta d z , the volume is then

Int(Int(Int(r,z = E .. ``),r = `` .. ``),theta = ``... = Int(Int(int(r,z = r .. sqrt(4-r^2)),r = D .. ``),th... = Int(Int([sqrt(4-r^2)-r]*r,r = D .. ``),theta = `` ....

= Int(Int([r*sqrt(4-r^2)-r^2],r = 0 .. sqrt(2)),theta... = 2*Pi*(8-4*sqrt(2))/3 .


As a check, execute Maple's built-in
TripleInt command from the student package. Keep in mind that, just as for the command is designed only for Cartesian-coordinate triple integrals. So you must give it the factor r in dV = r dr d theta dz as part of the integrand!

> with (student):
value(Tripleint(r, z = r..sqrt(4 - r^2), r = 0..sqrt(2), theta = 0..2*Pi) );

-8/3*sqrt(2)*Pi+16/3*Pi


2. More general surfaces and regions . Maple's cylinderplot command can plot any cylindrical-coordinate equation for which z or theta is expressible as a function of the remaining two variables. Recall that the syntax of the extended cylinderplot command requires entry of the formulas for all three of r , theta , and z (in that order), and specification of the range of the two independent ones, for which we use the generic symbols u and v:

cylinderplot( [r, theta , z], u = a..b, v = c..d ) ;


For example, the next routine plots the two simple cylindrical surfaces with equations
z = c and theta = kappa . (In the second plot, if you change to multiples of , the initial rendering appears edge-on, so click and rotate the plot slightly to see the plane.)

> with (plots):
cylinderplot( [r, theta, 3], r = 0..2, theta = 0..2*Pi, axes = boxed );
cylinderplot( [r, 2, z], r = 0..2, z = 0..3, axes = boxed );


The concluding example of this worksheet involves a triple integral over a region whose bounding surfaces are plotted by the extended cylinderplot command. It is representative of a common way that cylindrical-coordinate triple integrals arise: by transformation of a complicated Cartesian-coordinate triple integral.
Example 2 . Evaluate Int(Int(Int(1/sqrt(x^2+y^2),z = E .. ``),y = `` .. ... if E is the first-octant region between the graphs of x^2+y^2 = 2*x and z^2 = x^2+y^2 .

Solution . First, note that the first equation transforms to the simpler polar equation r^2 = 2*r*cos*theta , that is, r = 2*cos*theta (since the origin occurs with coordinates [0, /2]). The second one transforms to the pair of cylindrical-coordinate equations z = r and z = r . Since E lies in the first octant, only the first applies. The following routine plots E , with coordinate axes. Again, after executing the command rotate the initial plot to get a good view of E 's interior.

> with (plots):
surf1 := cylinderplot (2*cos(theta), theta = 0..Pi/2, z = 0..2, axes = boxed):
surf2 := cylinderplot ( [r, theta, r], r = 0..2, theta = 0..Pi/2):
xaxis := spacecurve([t, 0, 0, t = 0..2, color = magenta]) :
yaxis := spacecurve([0, t, 0, t = 0..2, color = magenta]) :
zaxis := spacecurve([0, 0, t, t = 0..2.5, color = magenta]) :
labx := textplot3d([2.1, 0, -.2, `x`], color = magenta):
laby := textplot3d([0, 2.1, -.2, `y`], color = magenta):
labz := textplot3d([0, .2, 2.2, `z`], color = magenta):
display(surf1, surf2, xaxis, yaxis, zaxis, labx, laby,labz);

[Maple Plot]


From the figure, the roof of the region is the conical surface, which an upward-directed arrow through
E hits as it emerges from the region, after entering E through the xy -plane. Thus, since dV = r dr d theta dz

Int(Int(Int(1/sqrt(x^2+y^2),z = E .. ``),y = `` .. ... = Int(Int(int(1,z = 0 .. r),r = D .. ``),theta = `` .... ,

where D is the half disk in the xy -plane with bounding curve r = 2 cos theta as theta varies from 0 to /2. Thus,


Int(Int(Int(1,z = E .. ``),y = `` .. ``),x = `` .. ... = Int(Int(r,r = 0 .. 2*cos*theta),theta = 0 .. Pi/2) = Int(2*cos^2*theta,theta = 0 .. Pi/2) .


Evaluate the last integral by means of the trigonometric identity
cos^2*theta = (1+cos(2*theta))/2 :

Int(2*cos^2*theta,theta = 0 .. Pi/2) = int([1+cos(2*theta)],theta = 0 .. Pi/2) = Pi/2 .


As before, you can check the entire calculation by using Maple's
TripleInt command from the student package:

> with (student):
value( Tripleint(1/sqrt(x^2 + y^2), z = 0..sqrt(x^2 + y^2), y = 0..sqrt(2 - x^2), x = 0..sqrt(2)) );

1/2*Pi