Question

7. F(x,y)=3xy, C is the portion of y=x^2 from (0,0) to
(2,4)

evaluate the line intergral (integral c f ds)

Answer #1

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

For the function
f(x,y)=x^3+y^3-3xy-ln(r)-e^c+t
You find that at the point (2,4) the value of the function is
50
f(2,4)=50
Suppose you were to estimate the values of the
following points:
A = （1.997，4.003）
B = （2.004，3.996）
C = （2.000，3.997）
D = （1.996，4.000）
At which point(s),would you expect (without calculation) the
value of the function be larger than f (2,4) = 50?
Also provide a quick (one line) explanation of why you would expect
the value to be larger than 50.

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.

Evaluate the following equation using interval arithmetic. (a)
f(x,y) = 3xy / x2−y+2 with the interval x = [.3, .4] and y = [.2,
.5]

Evaluate the vector line integral F*dr of F(x,y) = <xy,y>
along the line segment K from the point (2,0) to the point (0,2) in
the xy-plane

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).

Evaluate ∫_0,3^2,4▒〖(2y+x^2 )dx+(3x-y)dy along〗 The parabola
x=2t, y=t^2 +3 Straight lines from (0,3) to (2,3) A straight line
from (0,3) to (2,4)

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

Problem 10. Let F = <y, z − x, 0> and let S be the surface
z = 4 − x^2 − y^2 for z ≥ 0, oriented by outward-pointing normal
vectors.
a. Calculate curl(F).
b. Calculate Z Z S curl(F) · dS directly, i.e., evaluate it as a
surface integral.
c. Calculate Z Z S curl(F) · dS using Stokes’ Theorem, i.e.,
evaluate instead the line integral I ∂S F · ds.

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