Question

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

Answer #1

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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.

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)

Consider the vector field F = <2 x
y^3 , 3 x^2
y^2+sin y>. Compute
the line integral of this vector field along the quarter-circle,
center at the origin, above the x axis, going from the point (1 ,
0) to the point (0 , 1). HINT: Is there a potential?

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)

. Let C be the curve x2+y2=1 lying in the
plane z = 1. Let ?=(?−?)?̂+?? =
(a) Calculate ∇×?
(b) Calculate ∫?∙?? F · ds using a parametrization of C and a
chosen orientation for C.
(c) Write C = ∂S for a suitably chosen surface S and, applying
Stokes’ theorem, verify your answer in (b)
(d) Consider the sphere with radius √22 and center the origin.
Let S’ be the part of the sphere that is above the...

Consider the vector field below: F ⃗=〈2xy+y^2,x^2+2xy〉 Let C be
the circular arc of radius 1 starting at (1,0), oriented counter
clock wise, and ending at another point on the circle. Determine
the ending point so that the work done by F ⃗ in moving an object
along C is 1/2.

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.

A particle of mass m moves about a circle of radius R from the
origin center, under the action of an attractive force from the
coordinate point P (–R, 0) and inversely proportional to the square
of the distance.
Determine the work carried out by said force when the point is
transferred from A (R, 0) to B (0, R).

Consider the hemispherical shell of inner radius 3 and outer
radius 7. The mass density at any point P(x,y,z) is directly
proportional to the square of the distance from P to the
origin.
Set up an integral in an appropriate coordinate system for the
moment of inertia about the z-axis. Simplify your integrand as much
as possible, but do not evaluate the integral.

Let fx,y (x,y) = 3 e^-(x+y) for 0 < x <1/2y and y>0. a)
Find f x(x) and f y( y) . b) Write out the integral
necessary to find , Fx,y ( u v) . DO NOT EVALUATE THE INTEGRAL.

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