Ch17 Practice Test
Sketch the vector field F.
F(x, y) = (x - y)i + xj
Evaluate the line integral, where C is the given curve. ∫C xy4ds. C is the right half of the circle x2 + y2 = 4
oriented counterclockwise.
Evaluate the line integral ∫C xyz2 ds. C is the line segment from (-2, 6, 0) to (0, 7, 5).
Evaluate the line integral ∫C F · dr, where C is given by r(t), 0 ≤ t ≤ 1.
F(x, y) = ex-1i + xy j,
r(t) = t4i + t5j
Evaluate ∫ y2dx + x dy along the following paths.
(a) C = C1 is the line segment from (-13, -7) to (0, 6)
(b) C = C2 is the arc of the parabola x = 36 - y2 from (-13, -7) to (0, 6).
Determine whether or not F is a conservative vector field. If it is, find a function f such that F =
enter NONE.
F(x, y) = (excos(y))i + (exsin(y))j
F(x, y) = (sin(xy) + xcos(xy)y)i + (x2cos(xy))j
Consider F and C below.
F(x, y, z) = yz i + xz j + (xy + 10z)k: C is the line segment from (2, 0, -1) to (6, 4, 3)
(a) Find a function f such that F = f.
(b) Use part (a) to evaluate ∫C f · dr along the given curve C.
f. If it is not,
Find the work done by the force field F in moving an object from P to Q.
F(x, y) = e-yi - xe-yj,
P(0, 4), Q(4, 0)
Consider the force field and circle defined below.
F(x, y) = x2 i + xy j,
x2 + y2 = 144
Find the work done by the force field on a particle that moves once around the circle oriented in the clockwise
direction.
Find the work done by the force field F(x, y) = x i + (y + 7)j in moving an object along an arch of the cycloid
r(t) = (t − sin t)i + (2 − cos t)j, 0 ≤ t ≤ 2π.
Find the work done by the force field F(x, y) = xsin(y)i + y j on a particle that moves along the parabola y = x2 from (1, 1)
to (2, 4).
Consider F and C below.
F(x, y, z) = eyi + xeyj + (z+1)ezk , C: r(t) = t i + t2j + t3k, 0 ≤ t ≤ 1
(a) Find a function f such that F = f.
(b) Use part (a) to evaluate ∫C
f · dr along the given curve C.
Evaluate the line integral by the two following methods.
(x − y)dx + (x + y)dy, C is counterclockwise around the circle with center the origin and radius 3.
(a) directly
(b) using Green's Theorem
Evaluate the line integral by the two following methods.
xy dx + x2y3 dy, C is counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 3).
(a) directly
(b) using Green's Theorem
Evaluate the line integral by the two following methods: x dx + y dy, C consists of the line segments from (0, 1)
to (0, 0) and from (0, 0) to (1, 0) and the parabola y = 1 - x2 from (1, 0) to (0, 1).
(a) directly
(b) using Green's Theorem
Use Green's Theorem to evaluate the line integral along the given positively oriented curve.
∫C xy2 dx + 4x2y dy,
C is the triangle with vertices (0, 0), (2, 2), and (2, 4)
Use Green's Theorem to evaluate the line integral along the given positively oriented curve.
∫C y3dx - x3dy, C is the circle x2 + y2 = 4
Use Green's Theorem to evaluate ∫C F·dr. (Check the orientation of the curve before applying the theorem.)
F(x, y) = ‹y2cos(x), x2 + 2ysin(x)›,
C is the triangle from (0, 0) to (1, 3) to (1, 0) to (0, 0).
Evaluate 2y2dx + 6xy dy, where C is the boundary of the semiannular region D in the upper half-plane between the
circles x2 + y2 = 16 and x2 + y2 = 25.
If F(x, y) = (-yi + xj)/(x2 + y2), show that ∫F · dr = 2π for every positively oriented simple closed path that encloses the
origin.
Evaluate the surface integral: ∫∫S z dS
S is the surface x = y + 5z2, 0 ≤ y ≤ 1, 0 ≤ z ≤ 2.
Evaluate the surface integral:∫∫S y dS
S is the part of the paraboloid y = x2 + z2 that lies inside the cylinder x2 + z2 = 4.
Evaluate the surface integral:∫∫S y2 dS
S is the part of the sphere x2 + y2 + z2 = 100 that lies inside the cylinder x2 + y2 = 25
Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words,
find the flux of F across S. For closed surfaces, use the positive (outward) orientation.
F(x, y, z) = xzey i − xzey j + z k. S is the part of the plane x + y + z = 9 in the first octant and has downward
orientation.
For closed surfaces, use the positive (outward) orientation.
F(x, y, z) = x i - z j + y k. S is the part of the sphere x2 + y2 + z2 = 25 in the first octant, with orientation toward
the origin.
For closed surfaces, use the positive (outward) orientation.
F(x, y, z) = xz i + x j + y k. S is the hemisphere x2 + y2 + z2 = 25, y ≥ 0, oriented in the direction of the positive yaxis.
For closed surfaces, use the positive (outward) orientation.
F(x, y, z) = x i + y j + 7 k. S is the boundary of the region enclosed by the cylinder x2 + z2 = 1 and the planes y=0
and x+y = 5.
For closed surfaces, use the positive (outward) orientation.
F(x, y, z) = x2 i + y2 j + z2 k. S is the boundary of the solid half-cylinder 0 ≤ z ≤
, 0 ≤ x ≤ 2.
Use Stokes' Theorem to evaluate ∫∫S curl F · dS.
F(x,y,z) = 6y cos z i + exsin z j + xey k. S is the hemisphere x2 + y2 + z2 = 9, z ≤ 0, oriented upward.
Use Stokes' Theorem to evaluate ∫∫S curl F · dS.
F(x,y,z) = x2z2 i + y2z2 j + xyz k
S is the part of the paraboloid z = x2+y2 that lies inside the cylinder x2+y2 = 1, oriented upward.
Use Stokes' Theorem to evaluate ∫∫S curl F · dS.
F(x,y,z) = x2y3z i + sin(xyz) j + xyz k. S is the part of the cone y2 = x2+z2 that lies between the planes y = 0 and y
= 1, oriented in the direction of the positive y-axis.
Use Stokes' Theorem to evaluate ∫∫S curl F · dS. F(x,y,z) = exycos(z) i + x2z j + xy k.
S is the hemisphere
Use Stokes' Theorem to evaluate ∫C F · dr. C is oriented counterclockwise as
viewed from above. F(x,y,z) = e-x i + ex j + ez k
C is the boundary of the part of the plane 3x+y+3z = 3 in the first octant.
Use Stokes' Theorem to evaluate ∫C F · dr. C is oriented counterclockwise as
viewed from above.
F(x,y,z) = xy i + 2z j + 4y k. C is the curve of intersection of the plane x+z = 7
and the cylinder x2+y2 = 81.
Evaluate ∫ F · dr, where F(x, y, z) = -y2i + 2xj + 3z2k and C is the curve of the intersection
of the plane y + z = 3 and the cylinder x2 + y2 = 16. (Orient C to be counterclockwise
when viewed from above.)
Use the Divergence Theorem to calculate the surface integral ∫∫S F · dS; that is,
calculate the flux of F across S.
F(x,y,z) = 3xy2 i + xez j + z3 k. S is the surface of the solid bounded by the
cylinder y2 + z2 = 9 and the planes x = -1 and x = 3.
Use the Divergence Theorem to calculate the surface integral ∫∫S F · dS; that is,
calculate the flux of F across S.
F(x,y,z) = x4 i - x3z2 j + 4xy2z k. S is the surface of the solid bounded by the
cylinder x2+y2 = 4 and the planes z = 0 and z = x+2.
, oriented in the
direction of the
positive x-axis.
Evaluate ∫∫ F · dS, where F(x, y, z) = xyi + (y2 + ez)j + cos(xy)k and S is the
surface of the region E bounded by the parabolic cylinder z = 4 - x2 and the
planes z = 0, y = 0, and y + z = 6.