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Section 16.4 : Line Integrals of Vector Fields

  1. Evaluate CFdr where F(x,y)=2x2i+(y21)j and C is the portion of x225+y29=1 that is in the 1st, 4th and 3rd quadrant with the clockwise orientation.
  2. Evaluate CFdr where F(x,y)=xyi+(4x2y)j and C is the line segment from (4,3) to (7,0).
  3. Evaluate CFdr where F(x,y)=(x3y)i+(x2+7x)j and C is the portion of y=x3+2 from x=1 to x=2.
  4. Evaluate CFdr where F(x,y)=xyi+(1+x2)j and C is given by r(t)=e6ti+(4e2t)j for 2t0.
  5. Evaluate CFdr where F(x,y,z)=(3x3y)i+(y310)j+yzk and C is the line segment from (1,4,2) to (3,4,6).
  6. Evaluate CFdr where F(x,y,z)=(x+z)i+y3j+(1x)k and C is the portion of the spiral on the y-axis given by r(t)=cos(2t)itj+sin(2t)k for πt2π.
  7. Evaluate CFdr where F(x,y)=x2i+(y2x)j and C is the line segment from (2,4) to (0,4) followed by the line segment form (0,4) to (3,1).
  8. Evaluate CFdr where F(x,y)=xyi3j and C is the portion of x2+y216=1 in the 2nd quadrant with clockwise rotation followed by the line segment from (0,4) to (4,2). See the sketch below.
    This curve starts with the portion of ${{x}^{2}}+\frac{{{y}^{2}}}{4}=1$ starting at (-1,0) and ending at (0,4) followed by a line starting at (0,4) and ending at (4,-2).
  9. Evaluate CFdr where F(x,y)=xy2i+(2y+3x)j and C is the portion of x=y21 from y=2 to y=2 followed by the line segment from (3,2) to (0,0) which in turn is followed by the line segment from (0,0) to (3,2). See the sketch below.
    This curve starts with the portion of $x=y^{2}-1$ starting at (3,-2) and ending at (3,2).  This is followed by a line starting at (3,2) and ending at the origin.  The final portion of the curve is a line starting at the origin and ending at (3.-2).
  10. Evaluate CFdr where F(x,y)=(1y2)ixj for each of the following curves.
    1. C is the top half of the circle centered at the origin of radius 1 with the counter clockwise rotation.
    2. C is the bottom half of x2+y236=1 with clockwise rotation.
  11. Evaluate CFdr where F(x,y)=(x2+y+2)i+xyj for each of the following curves.
    1. C is the portion of y=x22 from x=3 to x=3.
    2. C is the line segment from (3,5) to (3,5).
  12. Evaluate CFdr where F(x,y)=y2i+(13x)j for each of the following curves.
    1. C is the line segment from (1,4) to (2,3).
    2. C is the line segment from (2,3) to (1,4).
  13. Evaluate CFdr where F(x,y)=2xi+(x+2y)j for each of the following curves.
    1. C is the portion of x216+y24=1 in the 1st quadrant with counter clockwise rotation.
    2. C is the portion of x216+y24=1 in the 1st quadrant with clockwise rotation.