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Exam 3- Take-Home Portion
Show all your work to receive full credit for a problem.
Attach this sheet to the solutions you hand in. Even if you attempt the problems
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I declare that the work I am submitting is entirely my own and that I did not confer with
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1. (12 pts) Evaluate the following line integrals. (NOTE: There may be more than one
way to figure out the answer).
(a)
ZC
2x
y2+ 1dx −2y(x2+ 1)
(y2+ 1)2dy
where Cis parametrized by x=t3−1, y =t6−t, 0≤t≤1.
(b)
IC
(x+z)dx +xdy +ydz
where Cis the upper half of the circle x2+z2= 9 in the plane y= 0, together
with the x-axis from (3,0,0) to (-3,0,0), oriented counterclockwise when viewed
from the positive y-axis.
2. Suppose Sis the solid given by
S={(x, y, z)|x2+y2+ 1 ≤z≤5}
and consider the vector field F=xi+yj+zk. Assume that ∂ S is oriented with an
outward normal.
(a) (6 pts) Compute the surface area of ∂S .
