Exercises

Identify the vector field F defined by the integrand, and then evaluate the line integrals over the given curve.

1. ó õ C  xdx,   2. ó õ C  ydx C: r( t) = á t,t2 ñ over [ 0,1] C: r( t) = át,t2 ñ over [ 0,1] 3. ó õ C  xdy 4. ó õ C  xdx C: r( t) = á et,1 ñ over [ 0,1] C: r( t) = áet,t2 ñ over [ 0,1] 5. ó õ C  xdy-ydx 6. ó õ C  xdy-ydx C: r( t) = á cos( t) ,sin( t) ñ , t in é ë 0,  p 2 ù û C: r( t) = t,   1-t2   over [ 0,1] 7. ó õ C  xdy-ydx 8. ó õ C  xdx+ydy C: r( t) = á cos( t) ,sin( t) ñ , t in [ 0,p] C: r( t) = á cos( t) ,sin(t) ñ ,  t in [ 0,2p] 9. ó õ C  xdx+ydy+zdz 10. ó õ C  xdx+ydy+zdz C: r( t) = á cos( t) ,sin( t) ,t ñ , t in [ 0,p] C : r( t) = á cos( t) ,sin( t) ,t ñ , t in [ 0,p] 11. ó õ C  yzdx+xzdy+xydz 12. ó õ C  yzdx+xzdy+xydz C: r( t) = á 3cos( t) ,5sin( t) ,4cos( t) ñ ,    t in [0,2p] C: r( t) = á 3cos( t) ,5sin( t) ,4cos( t) ñ,    t in [ 0,2p]

Find the work done by an object moving along the given curve through the given vector field.

13. F( x,y) = á y,-x ñ 14. F( x,y) = á y,-x ñ C: r( t) = á cos( t) ,sin( t) ñ , t in [ 0,p] C: r( t) = á et,e-t ñ , tin [ 0,1] 15. F( x,y) = áx2-y2,2xy ñ 16. F( x,y) = á x2-y2,2xy ñ C: r( t) = á cos( t) ,sin( t) ñ , t in [ 0,2p] C: r( t) = á et,e-t ñ , tin [ 0,1] 17. F( x,y,z) = áx2-y2,2xy,x2+y2 ñ 18. F(x,y,z) = á x2-y2,2xy,x2+y2 ñ C: r( t) = á t,t2,t3 ñ , t in [ 0,1] C: r( t) = á et,e-t,t ñ , t in [ 0,1] 19. F( x,y,z) = á x,y,z ñ 20. F( x,y,z) = á yz,zx,xy ñ C: r( t) = á cos( t) ,sin( t) ,t ñ , t in [ 0,4p] C: r( t) = á et,e-t,t ñ , t in [ 0,1]

Evaluate the following arclength line integrals:

21. ó õ C  x ds 22. ó õ C  xds C: r( t) = á t,t2 ñ , tin [ 0,1] C: r( t) = á sin( 2t) ,cos( t) ñ , t in [ 0,p] 23. ó õ C  ( x2+y2) z ds 24 ó õ C  (x2+yz) ds C: r( t) = á sin( 3t) ,cos( 3t) ,4t ñ , t in [ 0,p] C: r( t) = á sin( 2t) ,cos( 2t) ,t ñ , t in [ 0,p] 25. ó õ C  F ·T  ds, F = áy,-x ñ 26. ó õ C  F ·T  ds, F( x,y) = á y,-x ñ C: r( t) = á cos( t) ,sin( t) ñ , t in [ 0,p] C: r( t) = á et,e-t ñ , tin [ 0,1]

Let C = C₁ÈC₂, where C₁ and C₂ are as shown below.

Suppose now that the following integrals have the given values

31. A 1 kg object is propelled from the earth's surface with an initial velocity vector of v₀ = á 0, 49, 49 ñ m/sec . It's motion through space is modeled by

r( t) = v0t -   1 2 gt2,  t  in  [ 0,10]

where g = á 0, 0, 9.8 ñ . How much work is done in moving the object through the gravitational vector field F( x,y) = á 0, 0, -9.8 ñ ?

Exercises 31 and 32

32. A 1 kg object moves near the earth's surface with initial position r₀ = á 0,0 ñ and initial velocity v₀ = á 49, 49 ñ (see figure above). It traces out a curve in the xy-plane whose equation is given by

y = x-  x2 490 ,    x in [ 0,490]

The gravitational vector field is F( x,y) = á 0, 0, -9.8 ñ .

1. Show that p( t) = á 0, 490t, 490t-490t² ñ , t in [ 0,1] , parametrizes the path of the object. How much work is done in moving this object through the gravitational vector field?
2. Show that q( t) = á 0, 490sin²(t), 245sin²( 2t) ñ , t in [ 0,p/2] , parametrizes the path of the object. How much work is done in moving this object through the gravitational vector field?
3. Show that neither p' (0) nor q' ( 0) are equal to v₀ = á0, 49,49 ñ ? Why do we obtain the same value for the work in both (a) and (b) and in exercise 31?

33. A box with a mass of 1 slug is moved from the origin to the point ( 0,10  feet, 5  feet) through the gravitational force field F( x,y) = á 0,0,-32 ñ in units of slug-feet per sec².

1. How much work is done in moving the box along the curve C₁ parametrized by r( t) = á0,10t,5t ñ for t in [ 0,1] ?
2. How much work is done in moving the box along the curve C₂ parametrized by r( t) = á0,10t,5t² ñ for t in [ 0,1] ?
3. How much work is done in moving the box from the origin to (0,9,0) along the y-axis and then moving the box from (0,9,0) to ( 0,10,5) along the curve r(t) = á 0,t+9,5t ñ for t in [ 0,1] (i.e., along the curve C₃).

34. The acceleration vector field in the orbit plane of the earth-moon system is given by

F( x,y) =  -95194.14 x (x2+y2) 3/2 ,  -95194.14 y ( x2+y2)3/2

How much work is done in moving the moon from ( 238957,0) to ( 0,238957) along its approximately circular orbit

r( t) = á 238957cos( 0.22997t),238957sin( 0.22997t) ñ

where distances are in miles and time is in days?

35. If C₁ is parametrized by r₁( t) = á x₁( t) ,y₁( t) ñ for t in [ a₁,b₁] and C₂ is parametrized by r₂( t) = á x₂( t) ,y₂(t) ñ for t in [ a₂,b₂] , then their union C₁ÈC₂ is defined to be the curve parametrized by r( t) for t in [ -1,1] , where

r( t) = ì í î r1[ ( b1-a1) t+b1] if t  in  [ -1,0] r2[ ( b2-a2) t+a2] if t  in  (0,1]

1. What is r( -1) ? What is r(0) ? What is r( 1) ? Why would we call this curve the union of C₁ and C₂?
2. What point from C₂ is not included in C₁ÈC₂?
3. What is C₁ÈC₂ when C₁ is the curve parametrized by r₁( t) = á cos(t), sin(t) ñ for t in [ 0,p] and C₂ is the curve parametrized by

   r2( t) =   t,- 1-t2     for  t  in  [ -1,1]

36. Use the definition of C₁ÈC₂ in problem 35 to show that

ó õ C1ÈC2  F·dr = ó õ C1  F·dr+ ó õ C2  F·dr

37. Set up and numerically approximate the integral

ó õ C  ( x2+y2) ds

over the curve C with parametrization r( t) = á t,t² ñ over [ 0,1] .

38. Set up and numerically approximate the integral

ó õ C  F·i  ds

where i is the unit vector parallel to the positive x-axis, F( x,y) = á y,-x ñ , and C is the curve r( t) = á e^t, e^-t ñ, t in [ 0,1] .

39. Write to Learn: Write a short essay which uses (4) to prove that

ó õ -C  F·dr = - ó õ C  F·dr

40. Write to Learn: Write a short essay which uses (4) to show that if F and G are vector fields containing a curve C, then

ó õ C  ( F+G) ·dr = ó õ C  F·dr+ ó õ C  G·dr.

assuming that all the integrals exist.