• If you're seeing this message, it means we're having trouble loading external resources on our website. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript - [Voiceover] We've used the definite integral to find areas. What I want to do now is to see if we can use the definitive role to find an arc length.

What do I mean by that? Well, if I start at this point on the graph of a function, and if I were to go at this point right over here, not a straight line, we know already how to find the distance in the straight line but instead we want to find the distance along the curve.

If we lay a string along the curve, what would be the distance right over here?

### Arc length

That's what I'm talking about by arc length. What we could think about it is okay, that's going to be from x equals a to x equals b along this curve.

So how can we do it? Well, the one thing that integration, integral calculus is teaching us is that when we see something that's changing like this, what we could do is we can break it up into infinitely small parts. Infinitely small parts that we can approximately with things like lines and rectangles, and then we could take the infinite sum of those infinitely small parts. So let me break up my arc length. Let me break it up into infinitely small sections of arc length. Let me call each of those infinitely small sections of my arc length a, I guess I could say a length to differential, an arc length to differential, I call it ds. I'll draw it a much bigger that when I at least I conceptualize what a differential is, just so that we could see it. What do I mean by breaking it up into these ds's?

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Well, if that's the ds, and then let me do the others in other colors, that's another infinitely small change in my arc length.

Another infinitely small change in my arc length.In this section we are going to look at computing the arc length of a function. We can then approximate the curve by a series of straight lines connecting the points.

In other words, the exact length will be. We can then compute directly the length of the line segments as follows. However, using the definition of the definite integralthis is nothing more than.

This formula is. Note the difference in the derivative under the square root! This is one of the reasons why the second form is a little more convenient. Before we work any examples we need to make a small change in notation. Instead of having two formulas for the arc length of a function we are going to reduce it, in part, to a single formula. Note that we could drop the absolute value bars here since secant is positive in the range given. There is a very common mistake that students make in problems of this type.

While that can be done here it will lead to a messier integral for us to deal with. However, as also noted above, there will often be a significant difference in difficulty in the resulting integrals. All the simplification work above was just to put the root into a form that will allow us to do the integral.

They are easy enough to get however. Doing this gives. It can be evaluated however using the following substitution. So, we got the same answer as in the previous example. From a technical standpoint the integral in the previous example was not that difficult. It was just a Calculus I substitution. However, from a practical standpoint the integral was significantly more difficult than the integral we evaluated in Example 2. So, the moral of the story here is that we can use either formula provided we can get the function in the correct form of course however one will often be significantly easier to actually evaluate.

The derivative and root will then be. The first couple of examples ended up being fairly simple Calculus I substitutions. However, as this last example had shown we can end up with trig substitutions as well for these integrals.

Notes Quick Nav Download.Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment is also called rectification of a curve. The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases. A curve in the plane can be approximated by connecting a finite number of points on the curve using line segments to create a polygonal path. Since it is straightforward to calculate the length of each linear segment using the Pythagorean theorem in Euclidean space, for examplethe total length of the approximation can be found by summing the lengths of each linear segment; that approximation is known as the cumulative chordal distance.

If the curve is not already a polygonal path, using a progressively larger number of segments of smaller lengths will result in better approximations. The lengths of the successive approximations will not decrease and may keep increasing indefinitely, but for smooth curves they will tend to a finite limit as the lengths of the segments get arbitrarily small.

## 8.1: Arc Length

This means. In other words, the curve is always rectifiable.

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The definition of arc length of a smooth curve as the integral of the norm of the derivative is equivalent to the definition. A curve can be parameterized in infinitely many ways. Curves with closed-form solutions for arc length include the catenarycirclecycloidlogarithmic spiralparabolasemicubical parabola and straight line.

The lack of a closed form solution for the arc length of an elliptic and hyperbolic arc led to the development of the elliptic integrals. In most cases, including even simple curves, there are no closed-form solutions for arc length and numerical integration is necessary. Numerical integration of the arc length integral is usually very efficient.

For example, consider the problem of finding the length of a quarter of the unit circle by numerically integrating the arc length integral. The point Gauss—Kronrod rule estimate for this integral of 1. This means it is possible to evaluate this integral to almost machine precision with only 16 integrand evaluations.

The mapping that transforms from polar coordinates to rectangular coordinates is. The mapping that transforms from spherical coordinates to rectangular coordinates is.

A very similar calculation shows that the arc length of a curve expressed in cylindrical coordinates is. Arc lengths are denoted by ssince the Latin word for length or size is spatium. Two units of length, the nautical mile and the metre or kilometrewere originally defined so the lengths of arcs of great circles on the Earth's surface would be simply numerically related to the angles they subtend at its centre.

The lengths of the distance units were chosen to make the circumference of the Earth equal 40 kilometres, or 21 nautical miles. Those are the numbers of the corresponding angle units in one complete turn. Those definitions of the metre and the nautical mile have been superseded by more precise ones, but the original definitions are still accurate enough for conceptual purposes and some calculations.

For example, they imply that one kilometre is exactly 0. Using official modern definitions, one nautical mile is exactly 1. For much of the history of mathematicseven the greatest thinkers considered it impossible to compute the length of an irregular arc. Although Archimedes had pioneered a way of finding the area beneath a curve with his " method of exhaustion ", few believed it was even possible for curves to have definite lengths, as do straight lines. The first ground was broken in this field, as it often has been in calculusby approximation.

People began to inscribe polygons within the curves and compute the length of the sides for a somewhat accurate measurement of the length. By using more segments, and by decreasing the length of each segment, they were able to obtain a more and more accurate approximation. In the 17th century, the method of exhaustion led to the rectification by geometrical methods of several transcendental curves : the logarithmic spiral by Evangelista Torricelli in some sources say John Wallis in the sthe cycloid by Christopher Wren inand the catenary by Gottfried Leibniz in In this section, we use definite integrals to find the arc length of a curve.

We can think of arc length as the distance you would travel if you were walking along the path of the curve. Many real-world applications involve arc length. If a rocket is launched along a parabolic path, we might want to know how far the rocket travels. Or, if a curve on a map represents a road, we might want to know how far we have to drive to reach our destination. The techniques we use to find arc length can be extended to find the surface area of a surface of revolution, and we close the section with an examination of this concept.

Functions like this, which have continuous derivatives, are called smooth. This property comes up again in later chapters. We start by using line segments to approximate the length of the curve.

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Although it might seem logical to use either horizontal or vertical line segments, we want our line segments to approximate the curve as closely as possible.

To help us find the length of each line segment, we look at the change in vertical distance as well as the change in horizontal distance over each interval. Then the length of the line segment is given by. The following example shows how to apply the theorem. Round the answer to three decimal places.

Use the process from the previous example. Although it is nice to have a formula for calculating arc length, this particular theorem can generate expressions that are difficult to integrate. We study some techniques for integration in Introduction to Techniques of Integration.

In some cases, we may have to use a computer or calculator to approximate the value of the integral. Use a computer or calculator to approximate the value of the integral.

Math 2B. Calculus. Lecture 17. Arc Length, Review Integration Techniques

We have just seen how to approximate the length of a curve with line segments. The concepts we used to find the arc length of a curve can be extended to find the surface area of a surface of revolution.

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Surface area is the total area of the outer layer of an object. For objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces.This website uses cookies to ensure you get the best experience. Arc Length. Conic Sections Trigonometry.

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Chemical Reactions Chemical Properties. Arc Length Calculator Find the arc length of functions between intervals step-by-step. Correct Answer :. Let's Try Again :. Try to further simplify. Learning math takes practice, lots of practice. Just like running, it takes practice and dedication. If you want Math notebooks have been around for hundreds of years. You write down problems, solutions and notes to go back Sign In Sign in with Office Sign in with Facebook.

Join million happy users! Sign Up free of charge:. Join with Office Join with Facebook. Create my account.Using Calculus to find the length of a curve.

Please read about Derivatives and Integrals first. Imagine we want to find the length of a curve between two points. And the curve is smooth the derivative is continuous. First we break the curve into small lengths and use the Distance Between 2 Points formula on each length to come up with an approximate answer:.

We can write all those many lines in just one line using a Sum :. Maybe we can make a big spreadsheet, or write a program to do the calculations And now suddenly we are in a much better place, we don't need to add up lots of slices, we can calculate an exact answer if we can solve the differential and integral.

### Arc Length

So the arc length between 2 and 3 is 1. Well of course it is, but it's nice that we came up with the right answer! Larger values of a have less sag in the middle And "cosh" is the hyperbolic cosine function. The curve is symmetrical, so it is easier to work on just half of the catenary, from the center to an end at "b":.

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