MATH 2203 –Quiz 6 (Version 1) Solution
November 20, 2009
NAME________________________________
Let Fbe the vector …eld
F(x; y) = 32j
and let Cbe the parabolic curve y=x2beginning at the point (0;0) and
ending at the point (1;1).
Evaluate the line integral
ZC
Fdr.
You must include all details of your work.
Solution: The easiest way to do this is to use the Fundamental Theorem
of Line Integrals: The vector …eld F(x; y) = 32jis conservative and has
potential function f(x; y) = 32y. Thus
ZC
Fdr=ZC
rfdr=f(1;1) f(0;0) = 32 0 = 32.
Now we show how to do the integral using the de…nition of line integrals:
The curve Ccan be parameterized as
r(t) = ti+t2j
0t1
from which we see that
r0(t) = i+ 2tj
and
F(r(t)) r0(t) = (32j)(i+ 2tj) = 64t.
Thus ZC
Fdr=ZC
F(r(t)) r0(t)dt =Z1
0
64t dt =32.
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