Question

**Compute the line integral of f(x, y, z) = x 2 + y 2 −
cos(z) over the following paths:**

(a) the line segment from (0, 0, 0) to (3, 4, 5)

(b) the helical path → r (t) = cos(t) i + sin(t) j + t k from (1, 0, 0) to (1, 0, 2π)

Answer #1

1.) Let f ( x , y , z ) = x ^3 + y + z + sin ( x + z ) + e^( x
− y). Determine the line integral of f ( x , y , z ) with respect
to arc length over the line segment from (1, 0, 1) to (2, -1,
0)
2.) Letf ( x , y , z ) = x ^3 * y ^2 + y ^3 * z^...

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).

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)

(a) Is the vector field F = <e^(−x) cos y, e^(−x) sin y>
conservative?
(b) If so, find the associated potential function φ.
(c) Evaluate Integral C F*dr, where C is the straight line path
from (0, 0) to (2π, 2π).
(d) Write the expression for the line integral as a single
integral without using the fundamental theorem of calculus.

Compute the line integral with respect to arc length of the
function
f(x, y, z) = xy2
along the parametrized curve that is the line segment from
(1, 1, 1)
to
(2, 2, 2)
followed by the line segment from
(2, 2, 2)
to
(−6, 6, 1).

Compute the line integral with respect to arc length of the
function f(x, y, z) = xy2 along the parametrized curve
that is the line segment from (1, 1, 1) to (2, 2, 2) followed by
the line segment from (2, 2, 2) to (−9, 6, 3).

Compute the line integral with respect to arc length of the
function f(x, y, z) = xy^2 along the parametrized curve that is the
line segment from (1, 1, 1) to (2, 2, 2) followed by the line
segment from (2, 2, 2) to (−3, 6, 8).

Let F~ (x, y, z) = x cos(x 2 + y 2 − z 2 )~i + y cos(x 2 + y 2 −
z 2 )~j − z cos(x 2 + y 2 − z 2 ) ~k be the force acting on a
particle at location (x, y, z). Under this force field, the
particle is moved from the point P = (1, 1, 1) to Q = (0, 0, √ π).
What is the work done by...

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)

Quesiton: Compute the surface integral of f(x,y,z)=x^2
over z=sqrt(x^2+y^2), 0<=z<=1.

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