Practice Problems on Sections 13.2, 13.3

1. Evaluate the integral∫C

(√x3

+ 2y − 4)ds, where C is the curve given

by x(t) = 3t2, y(t) = 2 − t, 0 ≤ t ≤ 1.

2. Evaluate the integral∫C

(xy − z) ds, where C is the curve given by

x(t) = sin t, y(t) = − cos t, z(t) = 2t+ 1, 0 ≤ t ≤ π2.

3. Evaluate the integral∫Cx dx−y dy+2z dz, where C is the line segment

from the point (0,−1, 1) to the point (−2, 0, 4).

4. Evaluate the integral∫Cy dx + 2x dy, where C consists of the part of

the parabola y = 1 − x2 from (−1, 0) to (0, 1) and the part of the parabolay = x2 + 1 from (0, 1) to (1, 2).

5. Evaluate the integral∫CF ·dr, where F(x, y) = x2i−xyj and C is given

by r(t) = ti− t2j, 0 ≤ t ≤ 1.

6. Evaluate the integral∫CF · dr, where F(x, y, z) = zi− xj + k and C is

given by r(t) = sin ti− tj + cos tk, π ≤ t ≤ 2π.

7. Find the work done by the force field F(x, y, z) = yi + zj + xk inmoving an object along the straight line from P = (2,−1, 5) to Q = (0, 3, 2).

8. Show that the given vector field is conservative. Find a potentialfunction for F and use it to evaluate the integral

∫CF · dr along the given

curve C.

a) F(x, y) = (3x2y3 lnx+ x2y3)i + (3x3y2 lnx)j,and C is the part of the curve y = lnx from the point (e, 1) to the point(e2, 2).

b) F(x, y, z) =2x cos(πy2 )

zi− πx2 sin(πy2 )

2zj− x2 cos(πy2 )

z2k,

and C is the line segment from (1, 0, 1) to (2, 2, 2).

9. Find the work done by the force field F(x, y) = 6xy2i+6x2yj in moving

an object from (2, 0) to (0,−3) along the ellips x2

4+ y2

9= 1 in the clockwise

direction.

10. A thin wire is bent into the shape of a semicircle x2 + y2 = 1, x ≥ 0.If the density function is ρ(x, y) = x+ y2, find the mass of the wire.