Section 8.4
Arc Length and Surface Area

In Section 8.3 we devloped methods for computing the volume of solids contained within a shell. In this section we will focus on quantifying the arc length of a graph and the surface area of a shell.

We approximated accumulated volume as we filled a shell with volumes of cylinders. We will approximate surface area of shells with surface area of cones (actually, parts of cones).

Motivation

Figure 8.4.1 shows an outline of a football stadium being planned in a major city. The plan leads to project engineers needing to compute the arc length of a mathematical curve.

Figure 8.4.1.

Figure 8.4.2 tells the story of a city's beautification plan gone awry. The problem the plan created required project engineers to compute the area of a mathematical surface.

Figure 8.4.2. Unintended consequences of a city's beautification plan.

We will use calculus to build mathematical tools to address problems like those arising in Figures 8.4.1 and 8.4.2.

Arc Length of a Function's Graph Over an Interval

The idea of arc length of a function's graph over an interval is quite simple. Imagine covering the graph with a piece of string. Then stretch the string along the axis that holds the function's independent variable. The length of the string is the graph's arc length over that interval. Figure 8.4.3 illustrates this idea.

Figure 8.4.3. A graph's arc length over an interval is the length of string that covers the graph.

The arc length of the graph in Figure 8.4.3 over the interval $[0,4]$ is about 12.9 units.

While the idea of arc length is simple, it requires a bit of work to compute an arc length for a given function. The big idea is to accumulate the lengths of segments that approximate the graph over intervals of length $\Delta x$.

Figure 8.4.4 displays a graph of a function f. When you play the animation you see seven phases in conceptualizing a method for approximating the accumulated arc length of $f$'s graph as the value of x varies from -1.5 to 1.5.

Figure 8.4.4. Seven phases in conceptualizing accumulated arc length of $f$'s graph as the value of its independent variable varies over an interval.

The seven phases in Figure 8.4.4 unpack the idea of approximating arc length by using line segments that approximate the graph. The phases are:

1. The value of x varies by $dx$ through intervals of length $\Delta x$.
2. Highlight points on $f$'s graph above the beginning of each $\Delta x$-interval.
3. Highlight a displacement of $dx$ from each point as $dx$ varies through each $\Delta x$-interval.
4. Highlight $dy$, where $dy=r_f(\mathrm{left}(x))dx$.
5. The value of $ds$ is an approximation of the graph's arc length as $dx$ varies through a $\Delta x$-interval.
6. As x varies from -1.5, the approximate arc length from 1.5 to x is the total accumulation of approximate arc lengths over completed $\Delta x$-intervals plus the varying approximate arc length over the current $\Delta x$-interval.
7. Let $\Delta x$ have an infinitesimal value. Then approximate accumulated arc length is essentially exact accumulated arc length of $f$'s graph.

To calculate arc length of a functions graph from $t=a$ to $t=x$ as the value of x varies, all we need is the rate at which accumulated arc length varies with respect to x.

Once we have the rate of change of arc length with respect to x, we calculate arc length over that interval by integrating the arc length's rate over that interval.

Figure 8.4.5 shows a frozen image of approximate arc length on the graph of $f$ while $dx$ varies through a $\Delta x$-interval.

Figure 8.4.5. $ds$ is an approximation of the arc length of $f$'s graph through this partial $\Delta x$-interval.

The value of $ds$ in relation to values of $dx$ and $dy$ can be derived using Pythagoras' Theorem: $$\begin{align} ds^2&=dx^2+dy^2\\[1ex] ds&=\sqrt{dx^2+dy^2}\\[1ex] ds&=\sqrt{dx^2+r_f(\mathrm{left}(x))^2 dx^2}\\[1ex] ds&=\sqrt{\left(1+\left(r_f(\mathrm{left}(x))\right)^2\right)dx^2}\\[1ex] ds&=\sqrt{1+\left(r_f(\mathrm{left}(x))\right)^2}dx\\[1ex] \frac{ds}{dx}&=\sqrt{1+\left(r_f(\mathrm{left}(x))\right)^2} \end{align}$$

For infinitesimal values of $\Delta x$,

• $r_f(\mathrm{left}(x))$ is essentially equal to $r_f(x)$, the exact rate of change of $f$ with respect to x.
• $ds/dx$ is essentially equal to $r_s(x)$, the exact rate of change of arc length on the graph of f with respect to x.

Therefore, $r_s$, the exact rate of change of arc length of $f$'s graph with respect to its independent variable, is $r_s(x)=\sqrt{1+\left(r_f(x)\right)^2}$.

Reflection 8.4.1. How does the quotient of differentials $ds$ and $dx$ give an approximate rate of change of arc length of $f$'s graph with respect to x?

Reflection 8.4.2. Interpret the statement, "For infinitesimal values of $\Delta x$, $r_f(\mathrm{left}(x))$ is essentially equal to $r_f(x)$" in regard to Phase 7 of Figure 8.4.4.

Accumulated arc length on the graph of $f$ from $a$ to x is therefore $$\begin{align} \color{red}{\text{(Eq. 8.4.1)}}\qquad s_f(a,x)&=\int_a^x r_s(t)dt\\[1ex] &=\int_a^x \sqrt{1+\left(r_f(t)\right)^2}dt \end{align}$$

Example 1

Figure 8.4.6 shows the graph of the function $h$ defined as $$h(x)=2\frac{\sin x}{x+1}e^{-\cos x}$$for $x \ge 0$.

Figure 8.4.6. Graph of the function h for values of x greater than or equal to 0.

Our task is to define a function that gives the graph's arc length over any interval of its domain.

Solution

We know from above that the arc length of $h$'s graph over any interval $[a,x]$ is $s_h(a,x)=\int_a^x r_s(t) dt$, where $r_s(x)=\sqrt{1+\left(r_h(x)\right)^2}$. To define $s$ we therefore need to define $r_s$ and, by implication, $r_h$.

We can use GC's capabilities to define $h$'s arc length function $s_h$: $$\begin{align} s_h(a,x)&=\int_a^x r_s(t)dt\\[1ex] r_s(x)&=\sqrt{1+\left(r_h(x)\right)^2}\\[1ex] r_h(x)&=\frac{d}{dx}h(x) \end{align}$$

Figure 8.4.7 shows the graph of $y=s_h(a,x)$ as x varies from $a=1$.

Figure 8.4.7.

Exercise Set 8.4.1

Use GC to do the following for each of Exercises 1 through 4:

• Graph the function.
• Place a string along the graph between $x=a$ and $x=b$. Place the string on the x-axis to estimate the graph's arc length.
• Define the arc length's rate of change function with respect to the arc length's independent variable.
• Define the arc length function that will give the arc length of the function's graph over any interval $[a,x]$.
• Graph the function's arc length function.
• Use the function's arc length function to evaluate the arc length over the indicated interval.

1. Function f defined as $f(x)=e^{\cos(x)}$. Interval: $x=2 \text{ to }x=5$

2. Function g defined as $g(y)=\ln(y)$. Interval:  $y=1 \text{ to }y=4$
3. Function j defined as $j(x)=3\cos(10\sin(4x))$. Interval  $x=-4 \text{ to }x=4$.
4. Function k defined as $k(y)=\dfrac{y^2}{1+y^3}$. Interval: $y=0 \text{ to }y=5$.

Surface Area

The idea of a shell's surface area is somewhat analogous to the idea of a graph's arc length.

The shell's surface area is the area of the region you would get if you could unwrap the shell and place it in a plane without distorting its surface. The area of the planar region is the surface area of the shell.

Figure 8.4.8 illustrates this idea with a right circular cone and a part of the cone (called a frustum).

A right circular conic shell is one example of what is called a developable surface--a surface that can be unwrapped into a plane without distorting it. Developable surfaces can be quantified using geometry and algebra. Click here to see the derivation for surface area of a right cone. We will use this formula often, so please study it.

Figure 8.4.9 Shows three shells. Part (a), a right cone, is discussed above. Shells like these often can be treated with algebra and geometry to compute their surface areas.

Part (b), a paraboloid, is made by rotating $y=x^2$ around the y-axis. It is not a developable surface. You cannot unwrap it into a plane without distorting the surface. We will use the calculus you now know to compute surface areas of shells like this.

Part (c) is called Boy's Surface. Boy's Surface also is not developable. Defining Boy's Surface and computing its area requires function definitions and integration techniques that you will learn in Calculus III.

Figure 8.4.9. Three surfaces: (a) A circular cone, (b) a paraboloid, and (c) Boy's Surface.

The Plan

We will follow the general theme that to define an accumulation function $S$ (in this case, accumulating surface area) we integrate the accumulation's rate of change function $r_S$ over an interval whose length varies with the function's independent variable.

We therefore need to come up with a function $r_S$ whose values $r_S(x)$ give the rate of change of accumulating surface area as the function's independent variable x varies.

Once we have $r_S$, we can define $S$ as$$S(a,x)=\int_a^x r_S(t)dt$$

Approximate Surface Area of Shells of Revolution Using Conical Frusta

Figure 8.4.10 shows a surface created by revolving a graph around the x-axis. When you play the animation you see the surface being covered by sections of cones (called frusta, plural of frustum).

Figure 8.4.10. A shell's surface area approximated by conical frusta. The first pass shows frusta graphed as the value of x varies by $dx$ through large $\Delta x$-intervals. The next two show frusta as x varies by $dx$ through smaller $\Delta x$-intervals.

Rate of Change of a Shell's Surface Area

Figure 8.4.11 isolates the frustum that is growing in slant height as x varies by $dx$. Please understand that we show a visually large value of $ds$ as an aid to understanding the following discussion. But the discussion speaks of $ds$ as varying thorugh $\Delta x$-intervals of infinitisemal size.

Figure 8.4.11 focuses on three things:

1. The varying frustum approximates the shell's surface area over the $\Delta x$-interval through which the value of x is varying. The segment labeled $ds$ is the differential of length that approximates the arc length of the graph over that $\Delta x$-interval.
2. The second image removes the original surface to highlight the frustum that approximates the shell's surface over the current $\Delta x$-interval.
3. The third image shows that $f(x_0)$ is the frustum's radius at the beginning of the current $\Delta x$-interval.

Figure 8.4.11.

Figure 8.4.12 unwraps the frustum to make it easier to see the rate at which the frustum's area varies as $ds$ varies.

• We know from Section 8.3 that a circle's rate of change of enclosed area with respect to its radius has the same numerical value as its perimeter.
• The rate of change of enclosed area of any segment of the circle will have the same numerical value as the segment's perimeter. So $dS\doteq \left(2\pi f(x_0)\right)ds$.
• Earlier in this section, we concluded that $ds$, the differential approximation of a function's arc length as x varies over intervals of infinitesimal length, is $ds=\sqrt{1+\left(r_f(x)\right)^2}dx$.
• $r_S$, the rate of change function for the shell's surface area with respect to x, is therefore $$r_S(x)=2\pi f(x)\sqrt{1+\left(r_f(x)\right)^2}$$

Figure 8.4.12. The frustum unwapped into a circular segment.
$ds$ is the approximation to the function's accumulated arc length over the current $\Delta x$-interval.

It is important to understand the role played by $ds$, the approximate arc length of $f$'s graph as x varies by $dx$, and $2\pi f(x)$, the frustum's base perimeter. Look back and forth at Figures 8.4.11 and 8.4.12:

• $ds$ on the frustum becomes the amount by which the circle's radius varies in the unwrapped frustum.
• $2\pi f(x)$, the frustum's base perimeter, becomes the perimeter of the circular section made by the unwrapped frustum.

Reflection 8.4.6. Print Figure 8.4.12. Cut out the shaded region (including the statement "$Perimeter=2\pi f(x)$"). Pull the ends of the region you cut out together. You will form a frustum with slant height $ds$ and base perimter $2\pi f(x)$. Compare your frustum to Figure 8.4.11, part (c).

Computing Surface Area of a Shell of Revolution

We can now complete our plan for computing surface area of a shell of revolution.

We started by creating a shell by rotating the graph of a function $f$ around an axis. We thought of the shell's surface area as accumulating as the function's independent variable varies, creating an accumulation function $S$ whose value $S(a,x)$ gives accumulated surface area from $a$ to x.

We wanted to end with a function $S(a,x)=\int_a^x r_S(t)dt$, where $r_S(x)$ is the exact rate of change function for $S$.

To define the surface area function S, we needed to define $r_S$, the function that give the exact rate of change of a shell's surface area with respect to the function's independent variable. We defined $r_S$ as $$r_S(x)=2\pi f(x)\sqrt{1+\left(r_f(x)\right)^2}$$

We can now define the surface area function S as $$\begin{align}\color{red}{\text{(Eq. 8.4.2)}}\qquad S(a,x)&=\int_a^x r_S(t)dt\\[1ex] &=\int_a^x 2\pi f(t)\sqrt{1+\left(r_f(t)\right)^2}dt \end{align}$$

Example 2

Figure 8.4.14 shows the graph of $y=f(x)$ where $f(x)=0.5+\dfrac{x}{\sqrt{x+1}}+0.1\sin(10x)$.

Figure 8.4.15 shows the shell generated by rotating the graph of $f$ over the interval $[0,2]$ around the x-axis and the approximate accumulated surface area as x varies by $dx$ through $\Delta x$-intervals of length 0.01.

Figure 8.4.16 shows a GC file that applies the definition of $S$ given in Eq. 8.4.2 to the problem of modeling this surface area..

Figure 8.4.16.

Exercise Set 8.4.2

1. $f(x)=3+\sqrt{0.5+x}$ rotated around the x-axis, $0\le x\le 4$. What is $S(1, 1.5)$? What does this number mean?

2. $g(y)=4-y^2$ rotated around the y-axis, $0\le y \le 1.5$. (Note: Enter $x=g(y)$ to graph x as a function of y.) What is $S(0.5, 2.1)$? What does this number mean?

3. Let $h(x)=1+x^2$. Rotate $$\begin{align}y&=h(x),\, 0.5\le x\le 2\\[1ex]x&=0.5,\,1.25\le y\le 5\\[1ex]y&=5,\, 0.5\le x\le 2\end{align}$$ around the y-axis. (See below.) Be sure to include surface area of the inner tube and accumulated surface area on the top as your accumulated surface area's independent variable varies.

   What is the surface area of the entire shell?

   

4. $j(y)=4-y^2$ rotated around the x-axis, $0\le y \le 2$. (Graph $x=j(y)$; rotate that graph around the x-axis. Consider the surface in two parts.)
5. A cone can also be generated by revolving $f(x)=ax+b,\, 0\le x\le -b/a$ around either the x-axis or the $y-axis$. Use integration to show that you get the standard formula for surface area of a conic shell (not including the base): $\pi r s$, where r is the radius of the cone's base and $s$ is the cone's slant height.
6. With accumulating volume, you were free to choose your independent variable for the accumulation regardless of the function's independent variable. Are you free to choose the independent variable when representing accumulated sufrace area of a shell made by rotating a function's graph around an axis? Explain.
7. Parts (a) and (b) refer to shells created by the graphs of $f$ and $g$, shown below, rotated around an axis. The graphs intersect at $(3.4424,2.9625)$.
1. Rotate the graphs around the y-axis. Define a function, and any necessary auxiliary functions, whose value gives the accumulated surface area of the shell at any value of the accumulation function's independent variable.
2. Rotate the graphs around the x-axis. Define a function, and any necessary auxiliary functions, whose value gives the accumulated surface area of the shell at any value of the accumulation function's independent variable. Include the shell's base.



8. (Adapted from Briggs &Cochran, Exercise 6.6.35)

   The ratio of an object's surface area to its volume (SAV) is important in animal biology.

   Animals typically generate heat at a rate that is proportional to their volume and lose heat at a rate that is proportional to their surface area.

   Therefore, animals with a low SAV ratio (such as elephants) tend to retain heat, whereas animals with a high SAV ratio (such as children) lose heat relatively quickly.

   1. What is the SAV ratio of a cube with side lengths $a$?
   2. What is the SAV ratio of a ball with radius 5 inches?
   3. Compare the SAV ratios of two humans who are shaped similarly, but one is twice as tall as the other.
   4. Define f as $f(x)=\sin(x)+0.3\cos(30x)$. What is the SAV ratio of the shell created by revolving the region bounded by the graph of f from $x=1$ to $x=2$ around the x-axis? Include the ends of the solid when computing its surface area.