Question

Calculate the line integral I C <y,z,x> where C is the curve of intersection of the sphere given by equation (x − 1)^2 + (y − 1)^2 + z^ 2 = 4 and the plane given by equation x = 1 oriented counterclockwise when viewed from the positive x-axis.

Answer #1

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.

Use Stokes" Theorem to evaluate (F-dr where F(x, y, z)=(-y , x-z
, 0) and the surface S is the part of the paraboloid : z = 4- x^2 -
y^2 that lies above the xy-plane. Assume C is oriented
counterclockwise when viewed from above.

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 H C F · dr, if F(x, y, z) = yi + 2xj + yzk, and C is
the curve of intersection of the part of the paraboliod z = 1 − x 2
− y 2 in the first octant (x ≥ 0, y ≥ 0, z ≥ 0) with the coordinate
planes x = 0, y = 0 and z = 0, oriented counterclockwise when
viewed from above. The answer is pi/4+4/15

Use Stokes' Theorem to evaluate
∫
C
F · dr
where F = (x +
8z) i + (6x +
y) j + (7y −
z) k and C is the curve of
intersection of the plane x + 3y + z
= 24 with the coordinate planes.
(Assume that C is oriented counterclockwise as viewed from
above.) Please explain steps. Thank you:)

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

a) Find a parametric equation for a curve given as an
intersection of a sphere x^2 + y^2 + z^2 = 1 and a plane x + z = 1,
where 0 ≤ a ≤ 1.
b) Do the contour plot of the function f(x, y) = x 2 −y 2 . The
contour plot is a collection of several level curves drawn on the
same picture (be sure to include level curves for positive,
negative and zero value of...

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.

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