MATH 251: Calculus 3, SET8

15: Multiple Integrals

15.6: Triple Integrals [in Rectangular Coordinates]

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

1. [1037/4]

syms x y z
I = int(int(int(6*x*y, z, 0, x+y), x, y, 2*y), y, 0, 1)
I = 
I_appx = double(I)
I_appx = 4.6000

2. [1037/8]

syms x y z
I = expand( int(int(int(x*y*exp(z), z, 0, 2-x^2-y^2), y, 0, 1), x, 0, 1) )
I = 
I_appx = double(I)
I_appx = 0.4881

3. [1037/12]

syms x y z
I = int(int(int(sin(y), z, 0, x), y, 0, pi-x), x, 0, pi)
I = 
I_appx = double(I)
I_appx = 2.9348

4. [1037/14]

syms x y z
% Note x^2-1 = 1-x^2 yields 2*x^2 = 2, whence x = -1 or x = 1.
I = int(int(int(x-y, z, x^2-1, 1-x^2), y, 0, 2), x, -1, 1)
I = 
I_appx = double(I)
I_appx = -5.3333

5. [1038/20]

Here we use pseudo cylindrical coordinates (like polar coordinates on steroids). See Section 15.7.
syms r theta y
% Now x^2 + z^2 = 8 - x^2 - z^2 gives 2*R^2 = 8 or R = 2 > 0.
V = int(int(int(1*r, y, r^2, 8-r^2), r, 0, 2), theta, 0, 2*pi)
V = 
V_appx = double(V) % in cm^3
V_appx = 50.2655

6. [1038/26]

syms x y z
f = sqrt(x)*exp(x*y*z)
f = 
I = int(int(int(f, z, 0, 2), y, 0, 1), x, 0, 4)
I = 
I_appx = double(I)
I_appx = 146.8130

7. [1039/42]

syms x y z
delta = y
delta = y
m = int(int(int(delta, z, 0, 1-x-y), y, 0, 1-x), x, 0, 1)
m = 
CM = 1/m * int(int(int(delta*[x y z], z, 0, 1-x-y), y, 0, 1-x), x, 0, 1)
CM = 
m_appx = double(m)
m_appx = 0.0417
CM_appx = double(CM)
CM_appx = 1×3
0.2000 0.4000 0.2000

8. [1039/46]

Again, cylindrical coordinates render the needful. See Section 15.7.
syms h k r z theta
delta = k
delta = k
Iz = int(int(int(delta*(r^2) * r, z, r, h), r, 0, h), theta, 0, 2*pi)
Iz = 

9. [1039/48]

Got sphere? Use spherical coordinates. See Section 15.8.
syms phi rho theta
delta = rho, x = rho*sin(phi)*cos(theta), y = rho*sin(phi)*sin(theta), z = rho*cos(phi)
delta = ρ
x = 
y = 
z = 
m = int(int(int(delta * rho^2*sin(phi), rho, 0, 1), phi, 0, pi/2), theta, 0, 2*pi)
m = 
CM = 1/m * int(int(int(delta*[x y z] * rho^2*sin(phi), rho, 0, 1), phi, 0, pi/2), theta, 0, 2*pi)
CM = 
Iz = int(int(int(delta*(rho*sin(phi))^2 * rho^2*sin(phi), rho, 0, 1), phi, 0, pi/2), theta, 0, 2*pi)
Iz = 

10. [1039/50]

syms x y z
delta = x^2 + y^2
delta = 
m = int(int(int(delta, z, 0, sqrt(9-y^2)), x, 0, y/3), y, 0, 3)
m = 
CM = 1/m * int(int(int(delta*[x y z], z, 0, sqrt(9-y^2)), x, 0, y/3), y, 0, 3)
CM = 
Iz = int(int(int(delta*(x^2+y^2), z, 0, sqrt(9-y^2)), x, 0, y/3), y, 0, 3)
Iz = 
%
m_appx = double(m)
m_appx = 11.2000
CM_appx = double(CM)
CM_appx = 1×3
0.3747 2.2089 0.9375
Iz_appx = double(Iz)
Iz_appx = 59.7943