Question

Evaluate ∫c(1−xy/2)dS where C is the upper half of the circle centered at the origin with radius 1 with the clockwise rotation followed by the line segment from (1,0)to (3,0) which in turn is followed by the lower half of the circle centerd at the origin of radius 3 with clockwise rotation.

Answer #1

Problem 2. Let C be the circle of radius 100, centered at the
origin and positively oriented. The goal of this problem is to
compute Z C 1 z 2 − 3z + 2 dz.
(i) Decompose 1 z 2−3z+2 into its partial fractions.
(ii) Compute R C1 1 z−1 dz and R C2 1 z−2 dz, where C1 is the
circle of radius 1/4, centered at 1 and positively oriented, and C2
is the circle of radius 1/4, centered...

Evaluate the line integral R C (x 2 + y 2 ) ds where C is the
line segment from (1, 1) to (2, 5).

Evaluate the line integral, where C is the given
curve.
C
xeyz ds, C is the line segment from
(0, 0, 0) to (3, 4, 2)

Evaluate the line integral ∫_c x^2 +y^2 ds where C is the line
segment from (1,1) to (2,3).

Evaluate the line integral, where C is the given
curve.
∫C xyz2 ds, C is the line segment from
(-1,3,0) to (1,4,3)

2. Use Green’s Theorem to evaluate
R
C F · Tds, where C is the square with vertices
(0,0),(1,0),(1,1) and (0,1) in the xy plane, oriented
counter-clockwise, and F(x,y) =
hx 3 ,xyi. (Please give a numerical answer here.)

Describe the shape of the parametrically defined curve (eg,
upper half of circle with radius = 3, centered at (1, -2) ) and
sketch. x(t) = 5cos(t) + 5; y(t) = 5sin(t) – 2, -pi <= t <=
pi.

a) Find the parametric equations for the circle centered at
(1,0) of radius 2 generated clockwise starting from
(1+21/2 , 21/2). <---( one plus square
root 2, square root 2)
b) When given x(t) = tcost, y(t) = sint, 0 <_ t. Find dy/dx
as a function of t.
c) When given the parametric equations x(t) =
eatsin2*(pi)*t, y(t) = eatcos2*(pi)*t where a
is a real number. Find the arc length as a function of a for 0
<_ t...

Evaluate F · dr, where F(x, y) = <(xy), (3y^2)> and C is
the portion of the circle x^2 + y^2 = 4 from (0, 2) to (0, −2)
oriented counterclockwise in the xy-plane.

Let C be the circle with radius 1 and with center (−2,1), and
let f(x,y) be the square of the distance from the point (x,y) to
the origin.
Evaluate the integral ∫f(x,y)ds

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