Question

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).

Answer #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).

Evaluate the line integral F * dr where F = 〈2xy, x^2〉 and the
curve C is the trajectory of rt = 〈4t−3, t^2〉 for −1 ≤ t≤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^...

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 point)
Evaluate the line integral ∫CF⋅dr∫CF⋅dr, where F(x,y,z)=3xi+4yj-zk
and C is given by the vector function r(t)=〈sint,cost,t〉,
0≤t≤3π/2.

Evaluate the line integral ∫CF⋅dr, where F(x,y,z)=5xi+yj−2zk and
C is given by the vector function r(t)=〈sint,cost,t〉, 0≤t≤3π/2.

Evaluate the triple integral.
2 sin (2xy2z3) dV, where
B
B =
(x, y, z) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 2, 0 ≤ z ≤ 1

1.
if F(x,y) = ye^x + e^x. find the integral over C of F * dr when C
consists of the line segments 0,1) to (0,2) and (0,2) to
(4,2)

(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.

Let F = < 2?, −2? > and ?, the region bounded by ? = sin ?
and ? = 0, for 0 ≤ ? ≤ ? oriented counterclockwise.
a) Evaluate the line integral ∮ ? ∙ ??,
b) Find the same result using an appropriate area integral.

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