8.1 Arc Length

1-12. Find the length of the curve.

1. , .

2. , .

3. , .

4. , .

5. , .

6. from to .

7. , .

8. , , .

9. Find the length of the curve as goes to .

, .

http://matrix.skku.ac.kr/cal-lab/cal-8-1-9.html

(You may do this with Sage at http://math1.skku.ac.kr/.

Open resources at http://math1.skku. ac.kr/pub/)

k,x=var('k,x');

@interact

def _(y = input_box(tanh(k*x), label="y="), kappa=slider(0,10,0.05,default=1,label='k')):

plot(tanh(kappa*x), x,1,2, color='purple').show(aspect_ratio=1, xmin=1, xmax=2, ymin=0, ymax=1)

Answer : 1.

10. , .

11. Find the length of the curve when goes to 0,

. .

http://matrix.skku.ac.kr/cal-lab/cal-8-1-11.html

k,x=var('k,x');

@interact

def _(y = input_box(1/2*k*x^2, label="y="), kappa=slider(0,10,0.05,default=1,label='k')):

html('$$Arc Length=%s$$'%( lim(integral(sqrt(1+(k*x)^2),x,0,1),k=kappa) ))

plot(1/2*kappa*x^2, x,0,1).show(aspect_ratio=1, xmin=0, xmax=1, ymin=0, ymax=1)

Answer : 1.

12. Find the length of the curve, , , .

http://matrix.skku.ac.kr/cal-lab/cal-8-1-12.html

t=var('t')

parametric_plot(((cos(t))^3,(sin(t))^3),0,2*pi,rgbcolor=hue(0.6))

13.(a) Graph the curve

, .

(b) Compute the lengths of inscribed polygons with , , and sides. (Divide the interval into equal subintervals.) Illustrate by sketching these polygons.

(d) Use your calculator to find the length of the curve to four decimal places. Compare with the approximations in part (b). Have we introduced Simpson's Rule with ?