Exercises

Convert to cylindrical coordinates and evaluate:

1. ó õ 1 -1  ó õ Ö 1-x2    -Ö 1-x2 ó õ 2 0  z  dz dy dx 2. ó õ 1 0  ó õ Ö 1-x2    -Ö 1-x2 ó õ 2 0  z  dz dy dx 3. ó õ 1 -1  ó õ Ö 1-x2    -Ö 1-x2 ó õ 1 0  æ è x2+y2  + z ö ø dz dy dx 4. ó õ 1 -1  ó õ Ö 1-x2    -Ö 1-x2 ó õ 1 0  2z x2+y2   dz dy dx 5. ó õ 1 -1  ó õ Ö 1-x2    -Ö 1-x2 ó õ 1 0   dz dy dx x2+y2+1 6. ó õ 1 -1  ó õ 0    -Ö 1-x2 ó õ 1 0   z dz dy dx x2+y2+1 7. ó õ 1 0  ó õ x 0  ó õ | x| +1 0   dz  dy dx | x| +1 8. ó õ 1 0  ó õ x 0  ó õ Ö x2+y2    0  dz  dy dx x2+y2 9. ó õ 3 0  ó õ Ö 9-x2    0 ó õ x2+y2 0  (z+1) 2dz dy dx 10. ó õ 4 0  ó õ Ö 16-x2    0 ó õ Ö 16-x2-y2    0   z  dz dy dx

Evaluate the following triple integrals using spherical coordinates.

11. ó õ 1 -1  ó õ Ö 1-x2    -Ö 1-x2 ó õ Ö 1-x2-y2    -Ö 1-x2-y2  dz dy dx x2+y2+z2 12. ó õ 1 -1  ó õ Ö 1-x2    -Ö 1-x2 ó õ Ö 1-x2-y2    -Ö 1-x2-y2  z dz dy dx x2+y2+z2 13. ó õ 1 -1  ó õ Ö 1-x2    -Ö 1-x2 ó õ Ö 1-x2-y2    -Ö 1-x2-y2  x dz dy dx x2+y2+z2 14. ó õ 1 -1  ó õ Ö 1-x2    -Ö 1-x2 ó õ Ö 1-x2-y2    -Ö 1-x2-y2  y dz dy dx x2+y2+z2 15. ó õ 4 -4  ó õ Ö 16-x2    -Ö 16-x2 ó õ Ö 16-x2-y2    0  dz dy dx x2+y2 16. ó õ 4 0  ó õ Ö 16-x2    0 ó õ Ö 16-x2-y2    -Ö 16-x2-y2  dz dy dx y2+z2 17. ó õ 3 0  ó õ Ö 9-x2    0 ó õ Ö 9-x2-y2    -Ö 9-x2-y2  ( x2+y2) dz dy dx 18. ó õ 4 0  ó õ Ö 16-x2    0 ó õ Ö 16-x2-y2    0 x2 dz dy dx

Identify the solid, and then find its volume.

19. r = 0 to r = 1 20. r = 1 to r = 2 f = 0 to f = p f = 0 to f = p q = 0 to q = 2p q = 0 to q = 2p 21. r = 0 to r = 1 22. r = 0 to r = 1 f = 0 to f =  p 4 f = 0 to f = p q = 0 to q = 2p q = 0 to q = p 23. below x2+y2+z2 = 1 24. inside x2+y2 = 1 above x2+y2 = z2 between z = 0 and z = 1

The following are volume charge densities of charge clouds contained in a sphere of radius 1 meter. Calculate the total charge inside the sphere. Consider r₀ to be a constant.

25. r( x,y,z) = 2  C/m3 26. r(x,y,z) = 4  C/m3 27. r( x,y,z) = r0  x2+y2+z2   C/m3 28. r( x,y,z) =  r0 x2+y2+z2   C/m3 29. r( x,y,z) = r0 e - x2+y2+z2        C/m3 30. r( x,y,z) = r0  e-(x2+y2+z2) x2+y2+z2   C/m3

 

31. The solid cone between the xy-plane and the right circular cone ( z-1) ² = x²+y² has a volume charge density of

r( x,y,z) = 1-( x2+y2) z2

What is the total charge contained inside the solid cone?

32. Suppose that two concentric spheres of radius a and b, respectively, with b > a are centered at the origin, and suppose that the volume charge density between the two spheres is

r( r,f,q) =  r0( b-a) z2 ( x2+y2+z3) 5/2

with r₀ constant. What is the total charge between the two spheres?

33. A certain sphere of radius 1 meter centered at (0,0,1) has a mass density of

m( x,y,z) =  x2+y2+z2     kg m3

What is the mass of the sphere?

34. Suppose that the solid S is the ''spherical cap'' between x²+y²+z² = 2 and z = 1 if the mass density is

m( x,y,z) =  z ( x2+y2+z2) 3/2

35. What is the center of mass of the hemisphere x²+y²+z² = R² with z ³ 0 if the mass-density m of the hemisphere is constant?

36. What is the center of mass of the solid above x²+y² = z² and below x²+y²+z² = 1 if the mass-density m is constant?

37. In example 6 of section 6, it is shown that the gravitational potential between a mass m located at the point ( 0,0,r) and a sphere of radius R centered at the origin with a constant mass density m is given by

U = -Gm ó õ ó õ ó õ S   mdV x2 + y2 + ( z-r)2

where S is the sphere. Convert to triple integrals and evaluate for r > R to show that a sphere with uniform mass density has the same potential as a point mass, namely,

U =  -GmM r

38. What is the gravitational potential of a sphere of radius R with uniform mass-density if r < R (that is, when the satellite is inside the earth)?

39. Write to Learn: The right circular cone with height h and base with radius R is the solid below the plane z = h and above the cone R²z² = h²( x²+y²) . In a short essay, show that the cone corresponds to

f = tan-1 æ è  R h ö ø

and then use integration in spherical coordinates to find its volume.

40. Write to Learn: In a short essay, explain why if f (x,y,z) is a function only of the distance of a point (x,y,z) from the origin-that is, if

f( x,y,z) = f æ è x2 + y2 + z2   ö ø

for all ( x,y,z) -and if S is a sphere of radius R centered at the origin, then

ó õ ó õ ó õ S  f( x,y,z) dV = 4p ó õ R 0    f( r)   r2 dr