MATH 251: Calculus 3, SET8

15: Multiple Integrals

15.7: Triple Integrals [in Cylindrical Coordinates]

These problems are done with the CAS. See Hand Solutions for details.

1. [1043/4]

rectangular to cylindrical coordinate conversion
syms r x y z theta
% (a)
s = sym(sqrt(2)), Px = -s, Py = s, Pz = sym(1)
s = 
Px = 
Py = 
Pz = 1
r = norm([Px Py]), theta = sym(atan2(Py,Px))
r = 2
theta = 
P_rec = [Px Py Pz], P_cyl = [r theta Pz]
P_rec = 
P_cyl = 
% (b)
s = sym(2), Qx = s, Qy = s, Qz = s
s = 2
Qx = 2
Qy = 2
Qz = 2
r = norm([Qx Qy]), theta = sym(atan2(Qy,Qx))
r = 
theta = 
Q_rec = [Qx Qy Qz], Q_cyl = [r theta Qz]
Q_rec = 
Q_cyl = 

2. [1043/8]

In the xy-plane, the equation would represent a circle. In xyz-space, it represents a circular cylinder.
syms z theta
r = 2*sin(theta), x = r*cos(theta), y = r*sin(theta)
r = 
x = 
y = 
figure
fsurf(x,y,z, [0 2*pi -2 2], 'MeshDensity', 16)
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
zticks(-2:2)
title('SET8, 1043/8')

3. [1043/10]

Equations in rectangular equations are converted to those in cylindrical coordinates.
syms r x y z theta
a = 2*x^2 + 2*y^2 - z^2 == 4
a = 
a_cyl = simplify( subs(a, [x y], [r*cos(theta) r*sin(theta)]) )
a_cyl = 
b = 2*x - y + z == 1
b = 
b_cyl = subs(b, [x y], [r*cos(theta) r*sin(theta)])
b_cyl = 

4. [1043/12]

The solid in the first octant is bounded by a quarter-cone, a quarter-disk on top, and (hidden) triangular patches in the xz-plane and yz-plane.
syms r z theta
x = r*cos(theta), y = r*sin(theta), z = r, s = sym(2)
x = 
y = 
z = r
s = 2
figure
fsurf(x,y,z, [0 2 0 pi/2], 'MeshDensity', 10); hold on % quarter-cone on side
fsurf(x,y,s, 'r', [0 2 0 pi/2], 'MeshDensity', 10) % quarter-disk on top
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
xticks(0:2); yticks(0:2); zticks(0:2)
title('SET8, 1043/12'); view(121,16)

5. [1043/16]

This solid is bounded by a cone, cylinder, and disk. We compute its volume.
syms r z theta
x = r*cos(theta), y = r*sin(theta), s = sym(0)
x = 
y = 
s = 0
figure
fsurf(x,y,r, [0 2 0 2*pi], 'g', 'EdgeColor', 'none'); hold on % cone above
fsurf(x,y,s, [0 2 0 2*pi], 'b', 'EdgeColor', 'none'); hold on % disk on bottom
fsurf(2*cos(theta), 2*sin(theta), z, 'r', [0 2*pi 0 2], 'EdgeColor', 'none') % cylinder on side
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
xticks(-2:2); yticks(-2:2); zticks(0:2)
title('SET8, 1043/16')
alpha 0.5
%
V = int(int(int(1*r, z, 0, r), r, 0, 2), theta, 0, 2*pi)
V = 
V_appx = double(V) % in cm^3
V_appx = 16.7552

6. [1043/18]

syms r z theta
x = r*cos(theta), y = r*sin(theta), s = sym(4)
x = 
y = 
s = 4
figure
fsurf(x,y,r^2, [0 2 0 2*pi], 'g', 'EdgeColor', 'none'); hold on % paraboloid
fsurf(x,y,s, [0 2 0 2*pi], 'r', 'EdgeColor', 'none') % disk on top
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
xticks(-2:2); yticks(-2:2); zticks(0:4)
title('SET8, 1043/18')
%
I = int(int(int(z*r, z, r^2, 4), r, 0, 2), theta, 0, 2*pi)
I = 
I_appx = double(I)
I_appx = 67.0206

7. [1044/20]

syms r u z theta
x = r*cos(theta), y = r*sin(theta), s = sym(0)
x = 
y = 
s = 0
figure
fsurf(x,y,y+4, [1 4 0 2*pi], 'g', 'EdgeColor', 'none'); hold on % elliptical ring on top
fsurf(x,y,s, [1 4 0 2*pi], 'r', 'EdgeColor', 'none') % disk on bottom
fsurf(cos(theta), sin(theta), u*(sin(theta)+4), [0 2*pi 0 1], 'm', 'EdgeColor', 'none')
fsurf(4*cos(theta), 4*sin(theta), u*(4*sin(theta)+4), [0 2*pi 0 1], 'y', 'EdgeColor', 'none')
fplot3(cos(theta), sin(theta), sin(theta)+4, [0 2*pi], 'k')
fplot3(cos(theta), sin(theta), s, [0 2*pi], 'k')
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
xticks(-4:4:4); yticks(-4:4:4); zticks(0:4:16)
title('SET8, 1044/20'); view(113,37); alpha 0.5
%
I = int(int(int((x-y)*r, z, 0, y+4), r, 1, 4), theta, 0, 2*pi)
I = 
I_appx = double(I)
I_appx = -200.2765

8. [1044/24]

The volume of a solid is computed using cylindrical coordinates.
syms r phi theta; syms z positive
z_at_rim = solve(z + z^2 == 2, z) % and r = 1 there
z_at_rim = 1
figure
x = r*cos(theta), y = r*sin(theta), s = sym(sqrt(2))
x = 
y = 
s = 
fsurf(x,y,r^2, [0 1 0 2*pi], 'g', 'MeshDensity', 12); hold on % paraboloid
x = s*sin(phi)*cos(theta), y = s*sin(phi)*sin(theta)
x = 
y = 
fsurf(x,y,s*cos(phi), [0 pi/4 0 2*pi], 'r', 'MeshDensity', 12) % sphere
axis equal; axis([-1 1 -1 1 0 1.5])
xlabel('x'); ylabel('y'); zlabel('z')
xticks(-1:1); yticks(-1:1); zticks(0:0.5:1.5)
title('SET8, 1044/24')
%
V = int(int(int(1*r, z, r^2, sqrt(2-r^2)), r, 0, 1), theta, 0, 2*pi)
V = 
V_appx = double(V) % in cm^3
V_appx = 2.2587

9. [1044/28]

Looking ahead to Section 15.8, the mass of a solid spherical ball is computed using spherical coordinates.
syms a k phi rho theta positive
delta = k*rho*sin(phi)
delta = 
m = int(int(int(delta * rho^2*sin(phi), rho, 0, a), phi, 0, pi), theta, 0, 2*pi) % mass
m = 

10. [1044/30]

The integral is computed by switching to cylindrical coordinates.
syms r z theta
I = int(int(int(r*r, z, 0, 9-r^2), r, 0, 3), theta, 0, pi)
I = 
I_appx = double(I)
I_appx = 101.7876