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 ∫_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...

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

Using MatLab
2. Given the parametric equations x = t^3 - 3t, y = t^2-3:
(a) Find the points where the tangent line is horizontal or
vertical (indicate which in a text line)
(b) Plot the curve parametrized by these equations to
confirm.
(c) Note that the curve crosses itself at the origin. Find the
equation of both tangent lines.
(d) Find the length of the loop in the graph and the area
enclosed by the loop.
3. Use what...

1.) Let f(x,y) =x^2+y^3+sin(x^2+y^3). Determine the line
integral of f(x,y) with respect to arc length over the unit circle
centered at the origin (0, 0).
2.)
Let f ( x,y)=x^3+y+cos( x )+e^(x − y). Determine the line
integral of f(x,y) with respect to arc length over the line segment
from (-1, 0) to (1, -2)

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