Math 230, Chapter 15: Sample Problems
1. Given that the vector field F=3x2y−2y3+ 5, x3−6xy2+ 2y.
(a) Show that Fis conservative.
(b) Find ZC1
F·drif C1is the closed elliptic curve given by r(t) = h2 + 3cos t, −1 + 4 sin ti0≤
t≤2π.
(c) Without evaluating ZC2
F·drdirectly, explain in details, how you would evaluate the integral,
ZC2
F·dr, where C2is half the elliptic curve given by r(t) = h2 + 3cos t, −1 + 4sin ti0≤t≤π.
2. Evaluate ZC
(x2+y2+z2)ds where Cis described by r(t) = sin(t)i+ cos(t)j+ 8tkfor 0 ≤t≤1.
3. Compute ZC
F·drwhere Cis described by r(t) = cos(t)i+ sin(t)j+tkfor 0 ≤t≤πand
F(x, y, z) = xi+yj+−5zk
4. Evaluate ZC
F·drwhere Cis the parabolic arc given by r(t) = t, 4t−t2, t ∈[1,4] joining (1,3) to
(4,0) and F=yi−xj.
5. Evaluate ZC
F·drwhere F= 2xy i+x2jwhere C=C1 + C2+C3is pictured in Figure 1 below.
-
6
7@@R
7s
(8,5)
C1
C2C3
Figure 1
x
y
-
6
x4+y4= 1
Figure 2
x
y
6. (a) Use Green’s Theorem to evaluate the line integral RC2y3dx+ (x4+6y2x)dy where Cis the curve
given by the straight line from (0,1) to (0,0), the straight line from (0,0) to (1,0), followed by
the curve from (1,0) back to (0,1) along the curve x4+y4= 1 as shown in Figure 2 above.
(b) Use Green’s theorem to set up a line integral (dt) to find the area in the first quadrant bounded
by the curve r(t) = cos3t, sin3t,0≤t≤π
2.(Hint: In your integral, you may need the
parametric curve r(t) = (1 −t)j,0≤t≤1 and the parametric curve r(t) = ti,0≤t≤1.)
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