# Arc length.   Set up the integral required to find the length of from [0,1] analytically.  Use midpoint rule with at least 10 sub-intervals to APPROXIMATE the length of the curve.   You may...

1. Arc length.
• Set up the integral required to find the length of from [0,1] analytically.
• Use midpoint rule with at least 10 sub-intervals to APPROXIMATE the length of the curve.   You may use a spreadsheet.
• Determine the anti-derivative using calculus techniques, and use it to find the exact value, than approximate to three decimal places.

kalau | (Level 2) Adjunct Educator

Posted on

Set up the integral required to find the length of from [0,1] analytically.

****Since you did not provide a function, I can only write out the formulas for you.

-The arc length formula is the integral of ds .

L=\int_a^b ds

Such that in terms of dx  and dy :

ds =sqrt(1+((dy)/(dx))^2) dx

ds =sqrt(1+((dx)/(dy))^2) dy

Find the length of ...insert function here...  from [0,1].  Since your function isn't given, I cannot continue on with this problem.

• Use midpoint rule with at least 10 sub-intervals to APPROXIMATE the length of the curve.   You may use a spreadsheet.
•  (*I will choose to use the minimum 10 sub-intervals)
• The midpoint formula to approximate integrals is:

int_a^b f(x) dx ~~Delta x[f(x_1)+f(x_2)+f(x_3)+...+f(x_n)]

Where [a,b]=[0,1], and delta x is:

Delta x = (b-a)/n = (1-0)/10 = 1/10

We are using 10 n sub-intervals, from zero to one defined by [0,1]:

0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9, and 1.0

Then, our midpoints will be in between our sub-intervals.  Substituting into the formula, we will have:

int_0^1 f(x) dx ~~(1/10)[f(0.5)+f(1.5)+f(2.5)+...+f(9.5)]

You will need to evaluate the function at:  f(0.5), f(1.5), f(2.5)...f(9.5)

After you have found the values at their midpoints, you can substitute those values into the equation, add the values, and multiply the quantity by one-tenth.

Determine the anti-derivative using calculus techniques, and use it to

find the exact value, than approximate to three decimal places.

I do not know what type of function your question has given, so I cannot continue.