Question

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

(−6, 6, 1).

Answer #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).

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)

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^...

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)

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)

Compute the work done by the force F= <sin(x+y), xy,
(x^2)z> in moving an object along the trajectory 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, 5) when force is
measured in Newtons and distance in meters.

A) Use the arc length formula to find the length of the
curve
y = 2x − 1,
−2 ≤ x ≤ 1.
Check your answer by noting that the curve is a line segment and
calculating its length by the distance formula.
B) Find the average value fave of the
function f on the given interval.
fave =
C) Find the average value have of the
function h on the given interval.
h(x) = 9 cos4 x sin x, [0,...

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).

For each vector field F~ (x, y) = hP(x, y), Q(x, y)i, find a
function f(x, y) such that F~ (x, y) = ∇f(x, y) = h ∂f ∂x , ∂f ∂y i
by integrating P and Q with respect to the appropriate variables
and combining answers. Then use that potential function to directly
calculate the given line integral (via the Fundamental Theorem of
Line Integrals):
a) F~ 1(x, y) = hx 2 , y2 i Z C F~ 1...

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