Question

Evaluate the **line integral** where C is the curve
formed by the intersection of the cylinder **x^2 + y^2 =
9** and the plane **x + z = 5**, travelled
**CLOCKWISE** as viewed from the positive z-axis, and
v is the vector function **v = xyi + 2zj + 3yk**.

Answer #1

Use Stokes' Theorem to evaluate
C
F · dr
where C is oriented counterclockwise as viewed from
above.
F(x, y, z) = 5yi + xzj + (x + y)k,
C is the curve of intersection of the plane
z = y + 7
and the cylinder
x2 + y2 = 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

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.
z2dx +
x2dy +
y2dz,
C
C is the line segment from
(1, 0, 0) to (5, 1, 2)

Evaluate the line integral, where C is the given
curve.
C
xeyz ds, C is the line segment from
(0, 0, 0) to (3, 4, 2)

Evaluate the line integral, where C is the given
curve.
∫C xyz2 ds, C is the line segment from
(-1,3,0) to (1,4,3)

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 line integral R C (x 2 + y 2 ) ds where C is the
line segment from (1, 1) to (2, 5).

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