Use Green's theorem to evaluate [(x^5+2y)dx +(4x-6y^3) dy where C is the circle
x² + y² = 4.
Answers
Answer:
Step-by-step explanation:
he plane curve C described in this problem sheet is oriented counterclockwise.
1. Evaluate the line integral
I
C
(x
2
sin2 x − y
3
)dx + (y
2
cos2
y − y)dy
where C is the closed curve consisting of x + y = 0, x
2 + y
2 = 25 and y = x and lying in
the first and fourth quadrant.
2. Let a square R be enclosed by C and
I
C
(xy2 + x
3
sin3 x)dx + (x
2
y + 2x)dy = 6.
Find the area of the square.
3. Let C be a simple closed smooth curve and α be a real number. Suppose
I
C
(αex
y + e
x
)dx + (e
x + yey
)dy = 0.
Find α.
4. Let D be the region enclosed by a simple closed piecewise smooth curve C. Let F, Fx and
Fy be continuous on an open set containing D. Show that
∫ ∫
D
Fxdxdy =
I
C
F dy and ∫∫
D
Fydxdy = −
I
C
F dx.
5. Let C be the ellipse x
2 + xy + y
2 = 1. Evaluate H
C
(sin y + x
2
)dx + (x cos y + y
2
)dy.
6. Let D be the region enclosed by a simple closed smooth curve C. Show that
Area of D =
I
C
xdy = −
I
C
ydx.
7. Evaluate the area of the region enclosed by the simple closed curve x
2/3 + y
2/3 = 1.
8. Find the area between the ellipse x
2
9 +
y
2
4 = 1 and the circle x
2 + y
2 = 25.
9. Let f : [a, b] → R be a non-negative function such that its first derivative is continuous.
Suppose C is the boundary of the region bounded above by the graph of f, below by the
x-axis and on the sides by the lines x = a and x = b. Show that
∫ b
a
f(x)dx = −
I
C
ydx.
10. Let D be the region enclosed by the rays θ = a, θ = b and the curve r = f(θ). Use Green’s
theorem to derive the formula
A =
1
2
∫ b
a
r
2
dθ
for the area of D.