# 2.4 Arc Length of a Curve and Surface Area

### Learning Objectives

- Determine the length of a curve with equation between the two given points.
- Determine the length of a curve with equation between the two given points.
- Find the surface area of a surface of revolution.

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.

We begin by calculating the arc length of curves defined as functions of then we examine the same process for curves defined as functions of (The process is identical, with the roles of and reversed.) 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.

## Arc Length of the Curve *y=f(x)*

In previous applications of integration, we required the function to be integrable, or at most continuous. However, for calculating arc length we have a more stringent requirement for Here, we require to be differentiable, and furthermore we require its derivative, to be continuous. Functions like this, which have continuous derivatives, are called * smooth* . (This property comes up again in later chapters.)

Let be a smooth function defined over We want to calculate the length of the curve from the point to the point We start by using line segments to approximate the length of the curve. For let be a regular partition of Then, for construct a line segment from the point to the point 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. Figure 1 below depicts this construction for

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. Because we have used a regular partition, the change in horizontal distance over each interval is given by The change in vertical distance varies from interval to interval, though, so we use to represent the change in vertical distance over the interval as shown in Figure 2 below. Note that some (or all) may be negative.

By the Pythagorean theorem, the length of the line segment is We can also write this as Now, by the Mean Value Theorem, there is a point such that Then the length of the line segment is given by Adding up the lengths of all the line segments, we get

This is a Riemann sum. Taking the limit as we have

We summarize these findings in the following theorem.

### Arc Length for *y=f(x)*

Let be a smooth function over the interval Then the arc length of the portion of the graph of from the point to the point is given by

Note that we are integrating an expression involving so we need to be sure is integrable. This is why we require to be smooth. The following example shows how to apply the theorem.

### Calculating the Arc Length of a Function of *x*

Let Calculate the arc length of the graph of over the interval

#### Solution

We have so Then, the arc length is

Substitute Then, When then and when then Thus,

Let Calculate the arc length of the graph of over the interval

#### Answer

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 the subsequent chapters of this text. Before we know them, we may have to stop after setting up the integral.

### Setting Up the Integral for the Arc Length of a Function of *x*

Let Set up the integral for the arc length of the graph of over the interval

#### Solution

We have so Then the arc length is given by

Let Set up the integral for the arc length of the graph of over the interval

#### Answer

#### Hint

Use the process from the previous example.

## Arc Length of the Curve *x=g(y)*

We have just seen how to approximate the length of a curve with line segments. If we want to find the arc length of the graph of a function of we can repeat the same process, except we partition the *y*-axis instead of the *x*-axis. Figure 3 below shows a representative line segment.

Then the length of the line segment is which can also be written as If we now follow the analogous steps to what we have done before, we get a formula for arc length of a function

### Arc Length for *x=g(y)*

Let be a smooth function over an interval Then, the arc length of the graph of from the point to the point is given by

### Setting up the Integral for the Arc Length of a Function of y

Let Set up the integral for the arc length of the graph of over the interval

#### Solution

We have so Then the arc length is

Let Set up the integral for the arc length of the graph of over the interval

#### Answer

#### Hint

Use the process from the previous example.

## Area of a Surface of Revolution

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. 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. For curved surfaces, the situation is a little more complex. Let be a nonnegative smooth function over the interval We wish to find the surface area of the surface of revolution created by revolving the graph of around the *x*-axis as shown in the following figure.

As we have done many times before, we are going to partition the interval and approximate the surface area by surface areas of simpler shapes. We start by using line segments to approximate the curve, as we did earlier in this section. For let be a regular partition of Then, for construct a line segment from the point to the point Now, revolve these line segments around the *x*-axis to generate an approximation of the surface of revolution as shown in the following figure.

Notice that when each line segment is revolved around the axis, it produces a band. These bands are actually pieces of cones (think of an ice cream cone with the pointy end cut off). A piece of a cone like this is called a** frustum** of a cone.

To find the surface area of the band, we need to find the lateral surface area, of the frustum (the area of just the slanted outside surface of the frustum, not including the areas of the top or bottom faces). Let and be the radii of the wide end and the narrow end of the frustum, respectively, and let be the slant height of the frustum as shown in the following figure.

We know the lateral surface area of a cone is given by

where is the radius of the base of the cone and is the slant height (see the following figure).

Since a frustum can be thought of as a piece of a cone, the lateral surface area of the frustum is given by the lateral surface area of the whole cone less the lateral surface area of the smaller cone (the pointy tip) that was cut off (see the following figure).

The cross-sections of the small cone and the large cone are similar triangles, so we see that

Solving for we get

Then the lateral surface area (SA) of the frustum is

Let’s now use this formula to calculate the surface area of each of the bands formed by revolving the line segments around the *x*-axis. A representative band is shown in the following figure.

Note that the slant height of this frustum is just the length of the line segment used to generate it. So, applying the surface area formula, we have

Now, as we did in the development of the arc length formula, we apply the Mean Value Theorem to select such that This gives us

Furthermore, since is continuous, by the Intermediate Value Theorem, there is a point such that so we get

Then the approximate surface area of the whole surface of revolution is given by

This * almost* looks like a Riemann sum, except we have functions evaluated at two different points, and over the interval Although we do not examine the details here, it turns out that because is smooth, if we let the limit works the same as a Riemann sum even with the two different evaluation points. This makes sense intuitively. Both and are in the interval so it makes sense that as both and approach Those of you who are interested in the details should consult an advanced calculus text.

Taking the limit as we get

Let us now consider the surface obtained by rotating the same curve about the -axis assuming that to ensure that the curve is on one side of the axis of revolution. We can still use the approach as above, approximating the sought surface area with the sum of surface areas of the bands, it’s only that now the frustum bands are going to be formed by rotating the line segment connecting the points and on the curve about the -axis. This means that the parameters of the frustum are going to be , , while remains the same. Therefore, we get

where is the midpoint of and comes from the application of the Mean Value Theorem, as before. Analogously, to what we’ve done above, we then obtain

As with the arc length, we can now use a similar approach to derive formulas for the areas of surfaces obtained by revolving a curve with equation about the and -axes. These findings are summarized in the following theorem.

### Surface Area of a Surface of Revolution

Let be a smooth function over the interval

- If , , then the area of the surface obtained by revolving the graph of around the
*x*-axis is given by - If , then the area of the surface obtained by revolving the graph of around the
*y*-axis is given by

Similarly, let be a smooth function over the interval

- If , , then the area of the surface obtained by revolving the curve around the
*y*-axis is given by - If , then the area of the surface obtained by revolving the curve around the
*x*-axis is given by

### Calculating the Surface Area of a Surface of Revolution 1

Let over the interval Find the surface area of the surface generated by revolving the graph of around the *x*-axis.

#### Solution

The graph of and the surface of rotation are shown in the following figure.

We have Then, and Then,

Let Then, When and when This gives us

Let over the interval Find the surface area of the surface generated by revolving the graph of around the *x*-axis.

#### Answer

#### Hint

Use the process from the previous example.

### Calculating the Surface Area of a Surface of Revolution 2

Let Consider the portion of the curve where Find the surface area of the surface generated by revolving the graph of around the *y*-axis.

#### Solution

Notice that we are revolving the curve around the *y*-axis, and the interval is in terms of so we want to rewrite the function as a function of . We get The graph of and the surface of rotation are shown in the following figure.

We have so and Then

Let Then When and when Then

Let over the interval Find the surface area of the surface generated by revolving the graph of around the *y*-axis.

#### Answer

#### Hint

Use the process from the previous example.

### Key Concepts

- The arc length of a curve can be calculated using a definite integral.
- The arc length is first approximated using line segments, which generates a Riemann sum. Taking a limit then gives us the definite integral formula. The same process can be applied to functions of
- The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution.
- The integrals generated by both the arc length and surface area formulas are often difficult to evaluate.

## Key Equations

**Arc Length of a Function of***x*

**Arc Length of a Function of***y*

**Surface Area of a Function of***x*revolved about the*x*-axis

**Surface Area of a Function of***x*revolved about the*y*-axis

**Surface Area of a Function of***y*revolved about the*y*-axis

**Surface Area of a Function of***y*revolved about the*x*-axis

### Exercises

For the following exercises, find the length of the curve with the given equation over the specified interval.

** 1. **

#### Answer

** 2. **

** 3. **

#### Answer

** 4. ** Pick an arbitrary linear function over any interval of your choice Determine the length of the corresponding curve with calculus and then verify the answer is correct by using geometry.

** 5. ** Find the surface area of the surface generated when the curve from to revolves around the *x*-axis, as shown below.

#### Answer

** 6. ** Find the surface area of the surface generated when the curve from to revolves around the *y*-axis.

For the following exercises, find the lengths of the given curves. If you cannot evaluate the integral, leave your answer in the integral form (set up).

** 7. ** from

#### Answer

** 8. ** from

(Hint: rewrite the equation of the curve in the form .)

** 9. ** from

#### Answer

** 10. ** from to

** 11. [Set Up]** on to

#### Answer

** 12. ** from

** 13. ** from

#### Answer

** 14. ** from

** 15. ** from

#### Answer

10

** 16. [Set Up]** from

** 17. ** from to

#### Answer

** 18. ** from

** 19. ** from to

#### Answer

** 20. [Set Up] ** from to

** 21. [Set Up] ** from

#### Answer

** 22.** from to

** 23. [Set Up]** from to

#### Answer

** 24. [Set Up]** from to

** 25. [Set Up]** from

#### Answer

** 26. [Set Up]** from to

For the following exercises, find the surface area of the surface generated when the given curve revolves around the *x*-axis. If you cannot evaluate the integral, leave your answer in the integral form (set up).

** 27. ** from to

#### Answer

** 28. ** from to

** 29. ** from

#### Answer

** 30. [Set Up]** from

** 31. ** from

#### Answer

** 32. ** from

** 33. ** from

#### Answer

** 34. [Set Up]** from

For the following exercises, find the surface area of the surface generated when the given curve revolves around the *y*-axis. If you cannot evaluate the integral, leave your answer in the integral (set up) form.

** 35.** from

#### Answer

** 36.** from

** 37. ** from

#### Answer

** 38. [Set Up]** from to

** 39. ** from

#### Answer

** 40. [Set Up]** from to

** 41. [Set Up]** from to

#### Answer

** 42. [Set Up]** from to

** 43. ** The base of a lamp is constructed by revolving a quarter circle around the *y*-axis from to as shown below. Set up an integral for the surface area of the base of the lamp and compute it.

#### Answer

** 44.** A light bulb is a sphere with radius in. with the bottom sliced off to fit onto a cylinder of radius in. and hight in., as shown below. The sphere is cut off at the bottom to fit exactly onto the cylinder, so the radius of the cut is in. Find the surface area of the light bulb (not including the top or bottom of the cylinder).

** 45. [Set Up]** A lampshade is constructed by rotating a curve around the *y*-axis from to as shown below. Set up an integral for the amount of material you would need to construct this lampshade, that is, its surface area.

#### Answer

** 46. [Set Up]** An anchor drags behind a boat according to the function where represents the depth beneath the boat and is the horizontal distance to the anchor from the back of the boat. If the anchor is 23 ft below the boat, set up the integral to determine how much rope you have to pull to reach the anchor.

** 47. [Set Up]** You are building a bridge that will span 10 ft. You intend to add decorative rope in the shape of where is the distance in feet from one end of the bridge. Set up the integral to determine how much rope you need.

#### Answer

** 48. ** Find the arc length of the curve from to

(Hint: Recall trigonometric identities.)

## Glossary

- arc length
- the arc length of a curve can be thought of as the distance a person would travel along the path of the curve

- frustum
- a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base

- surface area
- the surface area of a solid is the total area of the outer layer of the 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

## Hint

Use the process from the previous example. Don’t forget to change the limits of integration.