#3: ARC LENGTH-SURFACE OF REVOLUTION

ARC LENGTH

You would use the Pythagorean Theorem to find the length of a straight line segment. What would you use if the curve was not a straight line? Again, you would slice the curve into small segments. The length of these small segment could be approximated by the length of the straight line connection the endpoints. (This line is called a chord.) Summing the lengths of the chords will approximate the length of the curve. The smaller the segments, the better the approximation until, in the limit, the corresponding integral will generate the length of the curve. This length is called an arc length.

1. Let y = x². Find the arc length of the curve from x = 0 to x = 2.

a) Graph the function.               b) Slice the curve into small segments

c) Look at a representative segment. Write the equation of the length of the chord, s, in terms of x and y.

d) Factor out x and approximate ( y )/( x ) = dy/dx

e) Set up the sum that would approximate the arc length.

f) Set up the integral that would calculate the arc length. Use technology to evaluate the integral.

2. HOMEWORK #1: You could have also solved for x in terms of y, factored out and approximated x/y by dx/dy. Set up the integral you would get if you did this.

3. Set up and evaluate the integral for the arc length of the following functions.

a) y = coshx from x = -2 to x = 2.

b) y = 2x^3/2 + 4 from x = 1 to x = 4.

SURFACES OF REVOLUTION

4. Go back to y = x². Revolve one of the s segments about the x-axis.

a) Describe the surface you get.               b) Write the geometric formula for this surface.

c) Write a formula for the surface area, S.

d) Set up the integral that would represent the surface area if this particular function from x = 0 to x = 2 is revolved about the x-axis.

e) Evaluate the integral.

f) Set up the integral that would represent the surface area if this particular function from x = 0 to x = 2 is revolved about the y-axis.

g) Evaluate the integral.

HOMEWORK ASSIGNMENT:

REQUIRED PROBLEMS	#1; Section 6.4: 6, 12, 14, 30
SUGGESTED PROBLEMS	Section 6.4: 7, 13, 17, 23, 27, 33, 35, 41, 47

Informative web sites:

• http://archives.math.utk.edu/visual.calculus/5/arclength.1/ Arc Length Tutorial by Larry Husch at University of Tennessee.
• http://www.ycef.com/virtual_classroom/math/calculus/index.html Animations for many solids of revolution by Yew Chung Education Foundation
• http://chaos.math.temple.edu/cgi-bin/manager Calculus on the web by Gerardo Mendoza and Dan Reich, Temple University. Go to book II, Applications of integration, Volumes or Assorted applications

READING ASSIGNMENT BEFORE NEXT WORKSHEET: Section 6.5