Question

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).

Answer #1

Evaluate double integral Z 2 0 Z 1 y/2 cos(x^2 )dx dy
(integral from 0 to 2)(integral from y/2 to 1) for cos(x^2) dx
dy

(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.

Evaluate Integral (subscript c) z dx + y dy − x dz, where the
curve C is given by c(t) = t i + sin t j + cost k for 0 ≤ t ≤
π.

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)

Evaluate C (y + 6 sin(x)) dx + (z2 + 2 cos(y)) dy + x3 dz where
C is the curve r(t) = sin(t), cos(t), sin(2t) , 0 ≤ t ≤ 2π. (Hint:
Observe that C lies on the surface z = 2xy.) C F · dr =

2. Is the vector field F = < z cos(y), −xz sin(y), x
cos(y)> conservative? Why or why not? If F is conservative, then
find its potential function.

Consider the following line integral of the conservative vector
ﬁeld: ZC(y2 sinz−z)dx + 2xy sinz dy + (xy2 cosz−x)dz where C is the
contour given by r(t) = ht3,2t2 −1,πti, 0 ≤ t ≤ 1/2. a. [4] Find
the potential f of the vector ﬁeld satisfying the condition
f(1,1,0) = 0. b. [5] Compute the line integral.

Evaluate the line integral of " (y^2)dx +
(x^2)dy " over the closed curve C which is the triangle
bounded by x = 0, x+y = 1, y = 0.

F(x, y, z) = sin y i + (x.cos y + cos z) j – y.sinz k
a) Determine whether or not the vector field is
conservative.
b) If it is conservative, find the function f such that F = ∇f
.

y = (6 +cos(x))^x
Use Logarithmic Differentiation to find dy/dx
dy/dx =
Type sin(x) for sin(x)sin(x) ,
cos(x) for cos(x)cos(x), and so on.
Use x^2 to square x, x^3 to cube
x, and so on.
Use ( sin(x) )^2 to square sin(x).
Use ln( ) for the natural logarithm.

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