Question

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.

Answer #1

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 =

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, where C is the given
curve.
xyeyz dy, C: x =
2t, y =
4t2, z =
3t3, 0 ≤ t ≤ 1
C

Evaluate the line integral, where C is the given
curve.
xyeyz dy, C: x =
2t, y =
2t2, z =
3t3, 0 ≤ t ≤ 1
C

Evaluate the line integral, where C is the given
curve.
xyeyz dy, C: x =
3t, y =
2t2, z =
4t3, 0 ≤ t ≤ 1
C

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.

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

Let w(x,y,z) = x^2+y^2+z^2 where x=sin(8t), y=cos(8t) , z=
e^t
Calculate dw/dt by first finding dx/dt, dy/dt, and dz/dt and using
the chain rule
dx/dt =
dy/dt=
dz/dt=
now using the chain rule calculate
dw/dt 0=

Evaluate ∫ C 2 xyz d x + x^2 z dy + x^2 y d z over the
path
c ( t ) = ( t^2 , sin ( π t /4 ) , e^(t^2 − 2t ) for 0
≤ t ≤ 2.

find dx/dx and dz/dy
z^3 y^4 - x^2 cos(2y-4z)=4z

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