Phys 2920, Spring 2009
Problem Set #7
1. (VA 5.28) If R(t) = (3t2
−t)i+ (2 −6t)j−4tk, find (a) RR(t)dt, and (b) R4
2R(t)dt
2. (VA 5.34) Evaluate R3
2A·dA
dt dt if A(2) = 2i−j+ 2kand A(3) = 4i−2j+ 3k.
3. (VA 5.38) If F= (5xy −6x2)i+ (2y−4x)j, evaluate RCA·dralong the curve Cin the
xy plane, y=x3from the point (1,1) to (2,8).
4. (VA 5.37) If A= (2y+ 3)i+xzj+ (y z −x)k, evaluate RCA·dralong the following paths
C:
(a) x= 2t2,y=t,z=t3from t= 0 to t= 1.
(b) the straight lines from (0,0,0) to (0,0,1), then to (0,1,1), and then to (2,1,1).
(c) the straight lines joining (0,0,0) and (2,1,1).
x
y
z
2
3
5. Evaluate the surface integral HSa·dS, where the vector
field is given (in cylindrical coordinates) by
a=ρ2cos2φˆ
eρ+ρsin φˆ
eφ+ρz3ˆ
ez
and the closed surface is a circular cylinder of radius 2
whose axis is the zaxis; it has height 3 and extends from
z= 0 to z= 3.
x
y
z
30o
6. Find the moment of inertia (about the zaxis) of
the “ice cream cone” volume which was used in another
example in class and which is shown here. (It is a sector
of a solid sphere of radius R, out to an angle θ=π/6
out from the zaxis)
Assume its mass density ρmass is uniform. Express
the answer in terms of the total mass Mof the object.
1