Arc length - applications of integrals, Mathematics

Arc Length - Applications of integrals

In this part we are going to look at determining the arc length of a function.  As it's sufficiently easy to derive the formulas that we'll utilize in this section we will derive one of them and leave the other to you to derive.

We want to find out the length of the continuous function

y = f (x) on the interval [a, b].

Primarily we'll need to find out the length of the curve. We'll do this by dividing the interval up into n equal subintervals each of width Δx and we'll indicate the point on the curve at each point by Pi. We can then estimate the curve by a series of straight lines connecting the points. Now Here is a sketch of this situation for n = 9.

132_Arc Length - Applications of integrals 4.png

Now indicate the length of every line segments by then be approximately, |Pi -1  Pi|  and the length of the curve will

206_Arc Length - Applications of integrals 3.png

and after that we can obtain the exact length by taking n larger and larger.  Alternatively, the exact length will be,

1974_Arc Length - Applications of integrals 2.png

Now here, let's get a good grasp on the length of each of these line segments. Very first, on each segment let's illustrate Δyi = yi - yi-1 = f (xi) - f (xi-1) . After that we can calculate directly the length of the line segments like this:

|Pi-1 Pi| = √ ((xi - xi-1)2 + (yi - yi-1)2)

= √(Δx2 +Δy2i).

By using the Mean Value Theorem we make out that on the interval [xi-1, xi] there is a point x*i that is why,

F (xi) - f (xi-1)

= f' (x*i) (xi - xi-1)

Δyi= f' (x*i)Δx

Hence, the length can now be written as,

|Pi-1 Pi| = √ ((xi - xi-1)2 + (yi - yi-1)2)

= √(Δx2 +[f' (xi*)]2 Δx2 )

= √ (1 + [f' (xi*)]Δx)

The exact length of the curve is then,

2388_Arc Length - Applications of integrals 1.png

Though, by using the definition of the definite integral, this is nothing much more than,

L - ∫ba√ (1+[f' (x)]2 dx)

A little more suitable notation (according to me) is the following.

L = ∫ba √ (1 + (dy/dx)2 dx)

In a identical way we can also derive a formula for x = h(y) on [c,d]. This formula is,

L - ∫bc√ (1+[h' (y)]2 dy)

bc √ (1 + (dx/dy)2 dy)

Once Again, the second form is possibly a much more convenient.

Note: the variation in the derivative under the square root! Don't get so confused. With one we distinguish with respect to x and with the other we distinguish with respect to y. One way to maintain the two straight is to note that the differential in the "denominator" of the derivative will match up along with the differential in the integral. This is one of the causes why the second form is a little much more suitable.

Previous to we work any instance we need to make a small change in notation. In place of having two formulas for the arc length of a function we are going to decrease it, in part, to a single formula. From this point on we are going to make use of the following formula for the length of the curve.

Posted Date: 4/12/2013 1:16:10 AM | Location : United States







Related Discussions:- Arc length - applications of integrals, Assignment Help, Ask Question on Arc length - applications of integrals, Get Answer, Expert's Help, Arc length - applications of integrals Discussions

Write discussion on Arc length - applications of integrals
Your posts are moderated
Related Questions
Class Mid points This is very significant values which mark the center of a provided class. They are acquired by adding together the two limits of a provided class and dividi

The product of -7ab and +3ab is (-7 x 3) a 2  b 2  = -21a 2  b 2 . In other words, a term with minus sign when multiplied with a term having a positive sign, gives a product having

I really need help with 30 60 90 right triangles and my last tutor did not make sense to me so can you please help

The following graph shows the growth of the median home value in a particular region of the United States starting in 1996.  The graphs starts in 1996 and shows the trend through t

Consider the equation x 2 y′′+ xy′- y = 4x ln x (a) Verify that x is a solution to the homogeneous equation. (b) Use the method of reduction of order to derive the second

Melissa is four times as old as Jim. Pat is 5 years older than Melissa. If Jim is y years old, how old is Pat? Start along with Jim's age, y, because he appears to be the young

Here, we have tried to present some of the different thinking and learning processes of preschool and primary school children, in the context of mathematics learning. We have speci

Cos(x+y)+sin(x+y)=dy/dx(solve this differential equation)

"Working" definition of continuity A function is continuous in an interval if we can draw the graph from beginning point to finish point without ever once picking up our penci

Explain how to Converting Percents to Decimals ? Percent : "Percent" means "per hundred." Percents are represented by a percent sign ( % ) to the right of a number.  For exam