13.3 Arc Length and Curvature

1-5. Find the length of the curve.


1. , .

 Sol) .

     .

     .


2. , .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-2.html

 Sol)

  

var('t')

r(t)=(2*sin(t)-t*cos(t),2*t,2*cos(t)+t*sin(t))

dr=diff(r(t),t);

s(t)=sqrt((dr[0]^2+dr[1]^2+dr[2]^2).simplify_trig());

length=integral(s(t),t,0,sqrt(5))

print length

  1/2*sqrt(5)*sqrt(10) + 5/2*arcsinh(1)


3. , .

 Sol) .

     .

     .


4. , .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-4.html

 Sol)

  

var('t')

x(t)=t

y(t)=2

z(t)=ln(t)

dx(t)=diff(x(t),t)

dy(t)=diff(y(t),t)

dz(t)=diff(z(t),t)

s(t)=sqrt(dx(t)^2+dy(t)^2+dz(t)^2)
length=integral(s(t),t,1,sqrt(3))

print length

  -sqrt(2) - 1/2*log(sqrt(2) - 1) + 1/2*log(sqrt(2) + 1) - 1/2*log(3) + 2


5. , .

 Sol) ,

      ,

      .


6-8. Reparametrize the curve with respect to arc length measured from the point where in the direction of increasing .


6. .

 Sol) Since , .

     Thus , this implies .

     Substituting in , we have .

     

7.

 Sol) Since , .

     Thus   .

     Substituting in , we have .


8. .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-8.html

 Sol)

  

var('t')

x(t)=3*t*sin(t)

y(t)=3*t*cos(t)

z(t)=--2*sqrt(2)*t^(3/2)

dx(t)=diff(x(t),t)

dy(t)=diff(y(t),t)

dz(t)=diff(z(t),t)

s(t)=sqrt(dx(t)^2+dy(t)^2+dz(t)^2)
length=integral(s(t),t,1,sqrt(3))

print length

3*sqrt(3)


9-10. Find the unit tangent , unit normal vectors and the curvature .


9. .

 Sol) , .

     Hence .

     , .

      Hence .

     .


10. , .


11-13. Use Theorem 7 to find the curvature.


11. .

 Sol) ,

     ,

     .


12. .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-12.html

 Sol)

  

var('t')

r(t)=(3+4*t^(3/2), 6*t,3/2*t^2)

dr=diff(r(t),t)

ddr=diff(r(t),t,2)

ANSWER=(dr.cross_product(ddr)).norm()/(dr.norm())^3

print ANSWER

  sqrt(abs(-9*sqrt(t))^2 + abs(-18/sqrt(t))^2 + 324)/(abs(3*t)^2 + abs(6*sqrt(t))^2 + 36)^(3/2)


13. .

 Sol) ,

     .

     .


14. Find the curvature of at the point .

 Sol) ,

     ,

          ,

     ,

     

                 ,

     , .


15. Find the curvature of at the point .

 Sol) ,

     ,

     ,

     .


16-18. Use Formula 6 to find the curvature.


16. .

 Sol) ,

     .


17. .

 Sol) ,

     .


18. .

 Sol) ,

     .


19-20. Find the vectors , and at the given point.


19. , .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-19.html

 Sol)

  

var('t')

r(t)=(t+1,2*t,t^2)

dr=diff(r(t),t)

T=dr/dr.norm()

dT=diff(T,t)

N=dT/dT.norm()

B=T.cross_product(N)

print T.subs(t=1)

print N.subs(t=1)

print B.subs(t=1)

  (1/3, 2/3, 2/3)

  (-2/15*sqrt(5), -4/15*sqrt(5), 1/3*sqrt(5))

  (2/5*sqrt(5), -1/5*sqrt(5), 0)


20. , .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-20.html

 Sol)

  

var('t')

r(t)=(exp(t)*cos(t),exp(t)*sin(t),sqrt(2)*exp(t))

dr=diff(r(t),t)

T=dr/dr.norm()

dT=diff(T,t)

N=dT/dT.norm()

B=T.cross_product(N)

print T.subs(t=0)

print N.subs(t=0)

print B.subs(t=0)

  (1/2, 1/2, 1/2*sqrt(2))

  (-sqrt(1/2), sqrt(1/2), 0)

  (-1/2*sqrt(1/2)*sqrt(2), -1/2*sqrt(1/2)*sqrt(2), sqrt(1/2))


21-22. Find equations of the normal plane and osculating plane of the curve at the given point.


21. .


http://matrix.skku.ac.kr/cal-lab/cal-13-3-21.html







 Sol)

  

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

r(t)=(2*sin(t),5*t,2*cos(t))

dr=diff(r(t),t)

T=dr/dr.norm()

dT=diff(T,t)

N=dT/dT.norm()

B=T.cross_product(N)

N=N.subs(t=pi)

expand(N[0]*(x-0)+N[1]*(y-5*pi)+N[2]*(z+2)==0)

  sqrt(1/29)*sqrt(29)*z + 2*sqrt(1/29)*sqrt(29) == 0


  

B=B.subs(t=pi)

expand(B[0]*(x-0)+B[1]*(y-5*pi)+B[2]*(z+2)==0)

 

  -10*pi*sqrt(1/29) + 5*sqrt(1/29)*x + 2*sqrt(1/29)*y == 0


22. .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-22.html

 Sol)

  

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

r(t)=(t^2, 2/3*t^3, t)

dr=diff(r(t),t)

T=dr/dr.norm()

dT=diff(T,t)

N=dT/dT.norm()

B=T.cross_product(N)

N=N.subs(t=1)

expand(N[0]*(x-1)+N[1]*(y-2/3)+N[2]*(z-1)==0)

  -1/3*x + 2/3*y - 2/3*z + 5/9 == 0


  

B=B.subs(t=1)

expand(B[0]*(x-1)+B[1]*(y-2/3)+B[2]*(z-1)==0)

 

  -2/3*x + 1/3*y + 2/3*z - 2/9 == 0


23. At what point on the curve is the tangent plane parallel to the plane

    .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-23.html

 Sol)

  

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

r(t)=(t^4, 3*t,t^2)

dr=diff(r(t),t)

T=dr/dr.norm()

T=T.subs(t=1)

expand(T[0]*(x-r(1)[0])+T[1]*(y-r(1)[1])+T[2]*(z-r(1)[2])==0)

  4/29*sqrt(29)*x + 3/29*sqrt(29)*y + 2/29*sqrt(29)*z - 15/29*sqrt(29) == 0