15.2 Line Integrals

1-4. Evaluate the following line integrals:


  1. Evaluate where C is the upper half of the unit circle .

http://math2.skku.ac.kr/home/pub/54 

 Sol)

 

t=var('t')

half_circle=parametric_plot((cos(t),sin(t)),(t,0,pi()));

show(half_circle, aspect_ratio=1)


 


 

integral(1+cos(t)^3*sin(t)^2*sqrt(diff(x,t)^2+diff(y,t)^2),t,0,pi())

  pi


2.    .

 Sol) Let . To evaulate the line integral, we used the Green's Theorem as following:

     


3. where is the arc of parabola in plane from to .

http://matrix.skku.ac.kr/cal-lab/cal-15-2-3.html

 Sol)

 

var('x,y,z')

p_1=implicit_plot3d(y==x^2, (x,0,1), (y, 0,1), (z, 0,4),color="red", opacity=0.2);

p_2=implicit_plot3d(z==2, (x,0,1), (y, 0,1), (z, 0,4),color="green");

show(p_1+p_2)

       

                     

 Using the Green's Theorem

   

integral(x^2*x^2+(x-2)*2,x,0,1);


4. where is the arc of circular helix from to .

 Sol) Let

     .


5. Determine whether the force field is conservative field.

http://matrix.skku.ac.kr/cal-lab/cal-14-2-5.html

 Sol)

 

var('x,y,z');

def curl(F):

    assert(len(F) == 3)

    return vector([diff(F[2],y)-diff(F[1],z), diff(F[0],z)-diff(F[2],x), diff(F[1],x)-diff(F[0],y)])

curl([2*x*z, x^2-y, 2*z-x^2])

  (0, 4*x, 2*x)


 Ans: , hence is not conservative field.


6. a. Prove that is a conservative field.

b. Find its scalar potential .

http://matrix.skku.ac.kr/cal-lab/cal-15-2-6.html

 Sol) scalar potential .

c. Also find the work done in moving an object in this field from to .

 Sol) work done.


      (Solution by SAGE)

 

x, y, z, i, j, k=var('x, y, z, i, j, k')

P(x,y,z)=4*x*y-3*x^2*z^2;

Q(x,y,z)=2*x^2;

R(x,y,z)=-2*x^3*z;

A(x,y,z)=P*i+Q*j+R*k;

A

  (x, y, z) |--> -2*k*x^3*z + 2*j*x^2 - (3*x^2*z^2 - 4*x*y)*i


 

curlA=(diff(R, y)-diff(Q,z))*i+(diff(P,z)-diff(R,x))*j+(diff(Q, x)-diff(P,y))*k;

curlA

  (x, y, z) |--> 0


 

f(x,y,z)=2*x^2*y-x^3*z^2

workdone=f(1,1,1)-f(0,0,0)

workdone

  1


7. If  .

    a. Prove that the line integral is independent of the curve joining two given points and .

    b. Show that there exists a scalar function such that and find .

    c. Also find the work done in moving an object from to .


8. Find the total work done in moving a particle in a force field along the curve form to .

http://matrix.skku.ac.kr/cal-lab/cal-15-2-8.html

 Sol)

 

var('t,x,y,z,F,A')

x(t)=t^2+1

y(t)=2*t^2

z(t)=t^3

r(t)=(x(t),y(t),z(t))

dr(t)=(diff(x(t),t),diff(y(t),t),diff(z(t),t))

A(t)=(3*x(t)*y(t),-5*z(t),10*x(t))

F(t)=A(t). dot_product (dr(t))

integral(F(t),t,1,2);

  303


9. Compute the area of the ellipse .

 Sol) Using the formula , we have  .


10. Find the area under of one arch of the asteroid , .


11. Find the area of the loop of the folium of Descartes , .

http://matrix.skku.ac.kr/cal-lab/cal-15-2-10.html 

 Sol)

 

@interact

def _(a=(1,(-10,20))):

 var('t')

 p_1=parametric_plot(((3*a*t)/(1+t^3),(3*a*t^2)/(1+t^3)), (t, 0,2*pi()), color="green");

 show(p_1)   

 def GreenThm(F):

    assert(len(F) == 2)

    return (diff(F[1],t)-diff(F[0],t))

 x=(3*a*t)/(1+t^3)

 y=(3*b*t^2)/(1+t^3)

 Anti=GreenThm((x,y))

 ANSWER=(1/2)*integral(Anti,t,0,2*pi())

 print ANSWER


 


12. Let . Find where is any smooth curve from to .

 Sol)

      .


13. Evaluate the line integral , where and is the curve given by , , .


 Sol) Let . We then observe that

      , . Note first that direct computations show that , which implies that there exists a scalar function with .

      We then have the following equations:

      , , .

      Solving these equations, one can see that , where is a constant.

      Thus, due to the fundamental theorem for line integral,

      .