Question

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^ 2 + sin( x + y
) * cos( x + z ). Determine the line integral of f ( x , y , z )
with respect to arc length over the curve r ( t ) = ( cos ( 2 π t )
, sin ( 2 π t ) , t ) where *t* ranges from 0 to 1

Answer #1

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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)

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 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)

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).

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).

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).

Let F ( x , y ) = 〈 e^x + y^2 − 3 , − e ^(− y) + 2 x y + 4 y 〉.
a) Determine if F ( x , y ) is a conservative vector field and, if
so, find a potential function for it. b) Calculate ∫ C F ⋅ d r
where C is the curve parameterized by r ( t ) = 〈 2 t , 4 t + sin
π...

Let F ( x , y , z ) =< e^z sin( y ) + 3x , e^x cos( z ) + 4y
, cos( x y ) + 5z >, and let S1 be the sphere x^2 + y^2 + z^2 =
4 oriented outwards Find the flux integral ∬ S1 (F) * dS. You may
with to use the Divergence Theorem.

Let F (x, y) =
2xyi + (x –
2y)j, r (t) =
sin ti – 2 cos t
j, 0 ≤ t ≤ π. Then C
F•dr is

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