Question

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)

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^...

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 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).

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?

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 = (sin(x 3 ), 2yex 2 ). Evaluate the line integral Z C F ·
dr, where C consists of two line segments, which go from (0, 0) to
(2, 2), and then from (2, 2) to (0, 2).

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.

double integral f(x,y)dA. f(x,y) = cos(x) +sin(y). D is the
triangle with vertices (-1,-2), (1,0) and (-1,2). You might notice
cos(x) is an even function, sin(y) is an odd function.

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