Exercises

Find the volume of the solid defined below.

1. f( x,y) = 1, g( x,y) = 0 2. f(x,y) = x+2, g( x,y) = 0 x = 0, x = 1, y = 0, y = 1 x = 1, x = 2, y = 0, y = 3 3. f( x,y) = xy, g( x,y) = 0 4. f(x,y) = x2+xy, g( x,y) = 0 y = 0, y = 1, x = y, x = 1 y = 0, y = 1, x = y, x = y2 5. f( x,y) = x+y, g( x,y) = x2+y2 6. f( x,y) = xy, g( x,y) = 4 x = 0, x = 1, y = 0, y = 1 y = 0, y = 1, x = y, x = 1

Find the mass of the solid defined below with the given mass density.

7. f( x,y) = xy, g( x,y) = 0 8. f(x,y) = x+2y, g( x,y) = 0 x = 0, x = 1, y = 0, y = 1 x = 1, x = 2, y = 0, y = 6 m( x,y,z) = 2 kg per cubic meter m(x,y,z) = 2 kg per cubic meter 9. f( x,y) = x2+y2, g( x,y) = 0 10. f( x,y) = x3+y2, g( x,y) = 0 y = 0, y = 1, x = y, x = 1 y = 1, y = 2, x = y, x = y2 m( x,y,z) = 2x kg per cubic meter m(x,y,z) = 2z kg per cubic meter 11. f( x,y) = x+y, g( x,y) = x2+y2 12. f( x,y) = xy, g( x,y) = 4 x = 0, x = 1, y = 0, y = 1 y = 0, y = 1, x = y, x = 1 m( x,y,z) = 2y kg per cubic meter m(x,y,z) = 2z kg per cubic meter

Find the total charge within the solid defined below with the given charge density ( m = meter).

13. f( x,y) = 1, g( x,y) = 0 14. f(x,y) = 4, g( x,y) = 2 x = 0, x = 1, y = 0, y = 1 x = 1, x = 2, y = 1, y = 6 r( x,y,z) = 2 coulombs per m3 r(x,y,z) = 5 coulombs per m3 15. f( x,y) = 1, g( x,y) = 0 16. f(x,y) = x3+y2, g( x,y) = 0 y = 0, y = 1, x = y, x = 1 y = 1, y = 2, x = y, x = y2 r( x,y,z) = 2z coulombs per m3 r(x,y,z) = yz coulombs per m3 17. f( x,y) = x+y, g( x,y) = x2+y2 18. f( x,y) = xy, g( x,y) = 4 x = 0, x = 1, y = 0, y = 1 y = 0, y = 1, x = y, x = 1 r( x,y,z) = x+y coulombs per m3 r(x,y,z) = 2x coulombs per m3

Find the center of mass of the solid defined below with the given mass density. (see 7-12 for the masses)

19. f( x,y) = xy, g( x,y) = 0 20. f(x,y) = x+2y, g( x,y) = 0 x = 0, x = 1, y = 0, y = 1 x = 1, x = 2, y = 0, y = 6 m( x,y,z) = 2 kg per cubic meter m(x,y,z) = 2 kg per cubic meter 21. f( x,y) = x2+y2, g( x,y) = 0 22. f( x,y) = x3+y2, g( x,y) = 0 y = 0, y = 1, x = y, x = 1 y = 1, y = 2, x = y, x = y2 m( x,y,z) = 2x kg per cubic meter m(x,y,z) = 2z kg per cubic meter 23. f( x,y) = x+y, g( x,y) = x2+y2 24. f( x,y) = xy, g( x,y) = 4 x = 0, x = 1, y = 0, y = 1 y = 0, y = 1, x = y, x = 1 m( x,y,z) = 2y kg per cubic meter m(x,y,z) = 2z kg per cubic meter

25. What is the volume of a right circular cylinder whose height is h and whose base has a radius of r?

26. What is the volume of a regular pyramid with a height h and a square base with sides of length s?

27. A certain type of concrete has a weight density at the earth's surface of 10 pounds per cubic foot. What is the weight of a concrete block in the shape of the solid

x = 0,x = 1,y = 0,y = 1,z = 0,z = y2

where all dimensions are in feet?

28. A mixture of sand and gravel is placed in a box with a square base of width 1 meter and a height of 2 meters. The box is then shaken vigorously causing more of the gravel to be near the bottom and more of the sand to be near the top.

What is the mass of the sand-gravel mixture in the box if it has a mass density of

29.  What is the center of mass of the tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1) if it has a uniform mass density of 36 kg per cubic meter?

30.  What is the mass of the pyramid with vertices (1,0,1), (1,0,-1), (-1,0,1), (-1,0,-1), and (0,2,0) if the mass density is r( x,y,z) = 3 - z kg per cubic meter?

31. A current density is a density of the form

j( x,y,z) = rm( x,y,z) v(x,y,z)

where r_m( x,y,z) is the density of the moving charges within a small section of a solid containing ( x,y,z) and v( x,y,z) is the velocity vector of a charge at ( x,y,z) if it is moving. What is the triple integral which represents the total current within the right circular cylinder

x2+z2 = R2

where R is a constant and where y is in [ 0,h] ?

32. Suppose that a solid S with mass dM produces a potential energy of

dU = -kmr  dM

on an object with mass m which is at a distance r from the small section. What is the triple integral form of the total potential energy when the solid has a mass density of m( x,y,z) ?

Exercises 33-39 deal with delta densities and point masses. Earlier exercises must be precede later exercises in this set.

33. The delta density d( x,y,z) is a density which satisfies

d( x,y,z) dV = ì í î 1 if ( 0,0,0)   is  in  S 0 if ( 0,0,0)   is  not  in  S

For example, the mass density of a solid S in which all of the mass M is ``concentrated'' at the origin is given by

m( x,y,z) = M  d( x,y,z)

What is the total mass of a solid S with such a mass density?

34. Suppose that all of the mass M of a solid S is ``concentrated'' at the point ( a,b,c) . Explain why the mass density for S is

m( x,y,z) = M  d( x-a,y-b,z-c)

What is the total mass of a solid S with such a mass density?

35. Suppose a solid S has a charge density of

r( x,y,z) = q1d( x,y,z) +q2d( x-a,y-b,z-c)

What does this density tell us about the charges inside of S? What is the total charge within S?

36. What is the center of mass of a solid S with a mass density of

m( x,y,z) = M1d( x,y,z) +M2d(x-a,y-b,z-c)

37. Write to Learn: In vector notation, we let r = á x,y,z ñ . Thus, if r₀ = áa,b,c ñ , then

d( x-a,y-b,z-c) = d( r-r0)

Write a short essay explaining why a collection of point masses m₁,¼,m_n located at points

r1,¼,rn

has a mass density of

m( r) = n å j = 1  mjd( r-rj)

and then find the total mass of the collection. Bonus: What is the center of mass of the collection?

38. Write to Learn: The delta density d( x,y,z) is more often defined by

d( x,y,z) f( x,y,z) dV = ì í î f( 0,0,0) if ( 0,0,0)   is  in  S 0 if ( 0,0,0)   is  not  in  S

when f( x,y,z) is a continuous function. Write a short essay in which you show that this definition reduces to the one in exercise 31 and that in addition, we have

d( x-a,y-b,z-c) f( x,y,z) dV = f(a,b,c)

when ( a,b,c) is in S.

39. Show that if we define

ó õ b a  d( x) dx = ì í î 1 if 0  is  in  [ a,b] 0 if 0  is  not  in  [ a,b]

then d( x,y,z) = d( x) d(y) d( z) . (i.e., show that it satisfies the definition in exercise 31).

40. Use Fubini's theorem for double integrals to prove Fubini's theorem for triple integrals, which says that

ó õ b a  ó õ d c  ó õ f e  f( x,y,z)dzdydx = ó õ d c  ó õ b a  ó õ f e  f( x,y,z) dzdxdy