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,y)-diff(F,z), diff(F,z)-diff(F,x), diff(F,x)-diff(F,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,t)-diff(F,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, .