Math 2210
Exam 4
Due Thursday, May 7, 2009
You are to work on this exam alone, if you have any questions, I will be around in the
mornings next week and also available by e-mail. In a pinch, you are welcome to call me
at home, 742-8695.
Show all work for complete credit.
(1) Evaluate the line integral RCxy4ds where Cis the lower half of the circle x2+y2=
16.
(2) Evaluate the line integral RCydx +xdy where Cis the straight line segment from
(0,0) to (1,2).
(3) Use a line integral to find the work done by the frce field F(x, y, z) = yzi+xzj+xy k
on a particle moving along the path r(t) = i+tj+t2k, 0 ≤t≤1.
(4) Show that Fis a conservative vector field and use that fact to find RCF·drwhere
F(x, y) = e2yi+ (1 + 2xe2y)jalong Cgiven by the lower semicircle from (−4,0) to
(1,0).
(5) Find the flux across the part of the surface z= 1 −x2−y2that lies above the
xy-plane by the vector field F(x, y, z) = yi+xj+zk.
(6) Use Green’s Theorem to evaluate RCeydx + 2xeydy where Cis the boundary of the
square with sides x= 0, x= 2, y= 0 and y= 2.
(7) Evaluate the surface integral R RSx2yzdS where Sis the part of the plane z=
6+2x+ 3ythat lies above the rectangle [0,2] ×[0,1].
(8) Use Stokes’ Theorem to find R RScurl(F)·dSgiven F(x, y , z) = x2eyzi+y2exz j+
z2exykwhere Sis the upper part of the hemisphere x2+y2+z2= 4, z≥0.
(9) Use the Divergence Theorem to find the flux of F(x, y, z ) = x2yi+xy2j+ 2xyzk
across Swhere Sis the surface of the tetrahedron defined by the planes x= 0,
y= 0, z= 0, and x+ 2y+z= 2
1