Question

Consider ∫_{c} x^{2}ydx + x^{3}dy where
C is the line segments going from (0,0) to (1,0) to (1,1) and back
to (0,0). Compute that path integral two ways: (a) directly via
parametrization, (b) using Green’s theorem.

Answer #1

Evaluate ∫C2x2+y2+1dS where C is the portion of the line
segments from the point (1,0) to (1,1) then from the point (1,1) to
(0,1) and then from the point (0,1) to (0,0).

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.)

2. Consider the line integral I C F · d r, where the vector
field F = x(cos(x 2 ) + y)i + 2y 3 (e y sin3 y + x 3/2 )j and C is
the closed curve in the first quadrant consisting of the curve y =
1 − x 3 and the coordinate axes x = 0 and y = 0, taken
anticlockwise.
(a) Use Green’s theorem to express the line integral in terms of
a double...

Consider a population with m+n elements, where n of them are 1
and m of them are 0. Then sample two points randomly from the
population.
a) Suppose we sample them without replacement, which corresponds
to the random sample. Calculate the probability of the outcomes
(0,0), (0,1), (1,0) and (1,1).
b) Suppose we sample them with replacement. Calculate the
probability of the outcomes (0,0), (0,1), (1,0) and (1,1).

In this and the following problem you will consider the
integral
∮? 6?sin(3?)??+6????
on the closed curve C consisting of the line segments
from (0,0) to (3,2) to (0,2) to (0,0). Here, you evaluate the line
integral along each of these segments separately (as you would have
before having attained a penetrating and insightful knowledge of
Green's Theorem), and in the following problem you will apply
Green's Theorem to find the same integral. Note that you can check
your answers...

Evaluate the integral ∬ ????, where ? is the square with
vertices (0,0),(1,1), (2,0), and (1,−1), by carrying out the
following steps:
a. sketch the original region of integration R in the xy-plane
and the new region S in the uv-plane using this variable change: ?
= ? + ?,? = ? − ?,
b. find the limits of integration for the new integral with
respect to u and v,
c. compute the Jacobian,
d. change variables and evaluate the...

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

Calculate ∮c(2?^2 − 3?) ?? + (?+ 2?^2)?? where C is a
closed curve (0,0) (2,0) (2,1)
a. With direct line integral
b. With Green Theory

a) How many grid paths from (0,0) to (6,6) are there?
b)
c) How many go through one of the points (1,1) or (2,2) or
(3,3)?
I need answer for C) please.
I'm trying to use the A + B + C - AnB - AnC - BnC
+ AnBnC rule (Principle of Inclusion/Exclusion) for this problem
but I found out that AnC(from 1,1 to 3,3 ) is just equal to
AnBnC(from 1,1 to 2,2 to 3,3) because for grid...

Problem 7. Consider the line integral Z C y sin x dx − cos x
dy.
a. Evaluate the line integral, assuming C is the line segment
from (0, 1) to (π, −1).
b. Show that the vector field F = <y sin x, − cos x> is
conservative, and find a potential function V (x, y).
c. Evaluate the line integral where C is any path from (π, −1)
to (0, 1).

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