Math 2263 Multivariable Calculus
Homework 26: 16.4 #8, 18
July 21, 2011
16.4#8
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.
Z
xe−2x dx + (x4 + 2x2 y 2 ) dy
C
C is the boundary of the region enclosed by the parabolas y = x2 and x = y 2 .
Green’s Theorem applies because C is the boundary of a closed region D, which is
most easily described in polar coordinates 1 ≤ r ≤ 2 and 0 ≤ θ ≤ 2π.
Z
ZZ ∂ 4
∂
2 2
−2x
(x + 2x y ) −
(xe ) dA
F · dr =
∂y
D ∂x
∂D
ZZ 3
2
4x + 4xy dA
=
D
ZZ =
4x(x + y ) dA
2
2
D
2π
Z
2
Z
4r cos θ(r2 )r dr dθ
=
0
1
2
Z
2π
Z
4r4 cos θ dθ dr
=
1
0
2
Z
2π
−4r sin θ dr
4
=
1
0
=0
16.4#18
A particle starts√at the point (−2, 0), moves along the x-axis to (2, 0), then along the
semicircle y = 4 − x4 to the starting point. Use Green’s Theorem to find the work done on
this particle by the force field F (x, y) = hx, x3 + 3xy 2 i.
We may use Green’s Theorem since C is a closed curve. If D is the region inside that
curve, we may most easily describe D in polar coordinates by 0 ≤ θ ≤ π and
0 ≤ r ≤ 2.
Z
ZZ ∂ 3
∂
3
2
2
hx, x + 3xy i · dr =
(x + 3xy ) −
(x) dA
∂y
C
D ∂x
ZZ 2
2
=
(3x + 3y ) − (0) dA
D
Z
π
Z
=
0
0
2
3r2 r dr dθ
Z
π
Z
2
3r3 dr dθ
=
0
0
0
2
3 4 r dθ
4 0
Z
π
π
Z
=
12 dθ
=
0
= 12π