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Section 8.3
Volumes and Surface Areas

Ancient civilizations were keen to find ways to compute volumes of solids. Archimedes (287-212 BCE) derived formulas for volumes of a cylinder, sphere, and cone. Chinese mathematicians in the same period derived formulas that approximated volumes of the same solids. Cavalieri (1598-1647) devised a method for computing volumes of solids that was a precursor of modern calculus. Computing volumes of solids was a testing ground for Newton's and Leibniz' new methods, grounded in the analysis of functions and change in functions' values, that launched the calculus.

Our treatment of volumes and surface areas of solids will build from Newton's approach. Newton thought of completed volumes and surface areas as having accumulated as they covaried with another quantity. As we did with areas of regions in the plane, we will build functions that model the accumulation of volume or surface area by determining the rate at which it varies at a moment as an independent variable varies through that moment.

In this section you will learn to conceptulize a solid by a two-step process:

  1. Think of the solid as an empty shell
  2. Fill the shell in a way that supports quantifying its volume.

Different Ways to Make a Solid's Shell

It is natural to look at a solid and to think that nothing about it changes--that it is rather than it is made. Figure 8.3.1, below, shows three solids like this. They appear as complete and unmade.


Figure 8.3.1. Examples of solids we will examine.

We will use these solids to exemplify the two step process mentioned above: (1) Think of the solid as an empty shell; (2) Fill the shell in a way that supports quantifing its volume.

Create a shell by varying a cross section

Figure 8.3.2 shows a quarter circle of radius 1 in the x-y plane. Run a square along an axis, perpendicular to the plane, so that the length of its sides is the perpendicular distance from the axis to the quarter circle. Do it again after covering the square's edges in pixie dust. The pixie dust leaves a record of everywhere each point on the square's edges has been, creating a shell that we will fill.


Figure 8.3.2 Start with a quarter circle in the x-y plane. A shell is formed by running a square of varying width perpendicularly to the plane along an axis.

Again, the length of the square's sides in Figure 8.3.2 at any place on the axis is the perpendicular distance from the axis to the quarter circle. The shell is created by sprinkling the square's edges with "pixie dust", so that the square leaves behind a trace of where it has been. The pixie dust has the effect of creating a shell that we will fill.

Create a shell by rotating a graph around the y-axis

Figure 8.3.3 shows the graph of $y=\sin(x).\, 0\le x \le \pi$. The animation first shows the graph rotated around the $y$-axis, then rotated again but sprinkled with "pixie dust". The pixie dust leaves a record of everywhere each point on the graph has been, creating a shell for the solid we will make by filling it.


Figure 8.3.3 Graph of $y=\sin(x)$ rotating around the $y$-axis, then rotated again sprinkled with pixie dust.

Create a shell by rotating a graph around the x-axis

Figure 8.3.3 shows the graph of $y=\sin(x).\, 0\le x \le \pi$. The animation first shows the graph rotated around the $x$-axis, then rotated again but sprinkled with "pixie dust". The pixie dust leaves a record of everywhere each point on the graph has been, creating a shell for the solid we will make by filling it.


Figure 8.3.4 Graph of $y=\sin(x)$ rotated around the $x$-axis, then rotated again sprinkled with pixie dust.

Different Ways to Fill a Shell

It is timely for us to talk about the idea of a cylinder, since we will "fill" each shape with cylinders that vary in height or radius.

A cylinder is a geometric shape that has these properties:

The volume of a cylinder is $V=A_bh$, where $V$ is the cylinder's volume, $A_b$ is the area of the cylinder's base, and $h$ is the cylinder's height measured perpendicularly from the plane containing the base.

The surface area of a cylinder's side is $A_s=Ph$, where $A_s$ is the area of the cylinder's side, $P$ is the perimeter of the cylinder's base, and $h$ is the cylinder's height.

A right cylinder is a cylinder whose sides are perpendicular to its top and bottom. We will use right cylinders exclusively to fill solids.

Figure 8.3.5, below, presents examples of right cylinders. All of them have congruent, parallel tops and bottoms with sides perpendicular to top and bottom. The volume of each is $V=A_bh$, where $V$ is the cylinder's volume, $A_b$ is the area of the cylinder's base, and $h$ is the cylinder's height measured perpendicularly from the plane containing the base.


Figure 8.3.5. Examples of right cylinders.

Reflection 8.3.1. Attend especially to (a) and (d) in Figure 8.3.5. Explain why they are cylinders.

Reflection 8.3.2. Explain why the formula $V=A_bh$ applies to the solid in Figure 8.3.1(d) even though it has a hole in it.

It will be useful to open Figure 8.3.1 in a new window while viewing the animations in this section. Think of each shape as a hollow shell that you will fill. The animations in this section suggest ways to fill these empty shells so that we may approximate their volumes with any degree of accuracy.

Fill a shell with cylinders, each with constant radius and varying height

Figure 8.3.6 shows the shell in part (a) of Figure 8.3.1. The animation shows it being filled with square cylinders of fixed width and varying height. As $h$ varies along its axis, each cylinder has constant width while its height varies by $dh$ through an interval of length $\Delta h$. A new cylinder begins when the value of $h$ passes into the next $\Delta h$-interval.


Figure 8.3.6 The shell is filled with square cylinders of fixed width and varying height.

We use "fill" in the sense of "filling a jar with marbles". There are gaps within the shell that are empty. So the volume of our fills approximates the volume of the solid within the shell. Smaller values of $\Delta h$ produce smaller gaps and therefore produce more accurate approximations to the exact volume.

Figure 8.3.7 shows the shell in part (b) of Figure 8.3.1. The animation shows it being filled with circular cylinders of fixed radius and varying height. As $h$ varies along its axis, each cylinder has constant radius while its height varies by $dh$ through an interval of length $\Delta h$. A new cylinder begins when the value of $h$ passes into the next $\Delta h$-interval.


Figure 8.3.7 The shell is filled with circular cylinders of fixed radius and varying height.

Figure 8.3.8 shows the shell in part (c) of Figure 8.3.1. The animation shows it being filled with circular cylinders of fixed radius and varying height. As $h$ varies along its axis, each cylinder has constant radius while its height varies by $dh$ through an interval of length $\Delta h$. A new cylinder begins when the value of $h$ passes into the next $\Delta h$-interval.


Figure 8.3.8 The shell is filled with circular cylinders of fixed radius and varying height.

Reflection 8.3.3 We used the word "height" for cylinders in Figures 8.3.6, 8.3.7 and 8.3.8. But the shell in Figures 8.3.6 and 8.3.8 is filled with cylinders that appear sideways. Why is it appropriate to say that the cylinders in Figures 8.3.6 and 8.3.8 have varying heights?

Fill a shell with cylinders, each with constant height and varying radius

Figure 8.3.9 shows the shell in part (b) of Figure 8.3.1. This time, the animation shows the shell being filled with circular cylinders of fixed height and varying radius. As $r$ varies along its axis, each cylinder has constant height while its radius varies by $dr$ through an interval of length $\Delta r$. A new cylinder begins when the value of $r$ passes into the next $\Delta r$-interval.


Figure 8.3.9 The shell is filled with circular cylinders of fixed radius and varying height.

Reflection 8.3.4. Figures 8.3.7 and 8.3.9 show the same shell being filled in two different ways. Describe how the ways of filling the same shell differ from each other yet produce essentially the same volume for sufficiently small values of $\Delta h$ and $\Delta r$.

Exercise Set 8.3.1

    The purpose of these exercises is for you to practice two things: conceptualize a shell and how it is made, and conceptualize a shell being filled by cylinders. You will do this by trying to imagine them and then sketch what you imagine.

    You will become better at visualizing shells and fills by attempting to sketch them, no matter how well you do it. You will not become better at it by looking quickly at hints and solutions before you have made a serious effort.

  1. The diagram below shows a leaf drawn in the x-y plane of a 3d-coordinate system. Imagine a semi-circle perpendicular to the x-axis and the plane. The semi-circle's lower corners lie on the edges of the leaf.

  2. The diagram below shows the same leaf as in #1. Imagine a rectangle perpendicular to the x-axis and the plane. The rectangle's lower corners lie on the edges of the leaf. Its height is half its width.

  3. The figure below shows a graph in the x-y plane. Imagine it rotated around the $y$-axis to make a shell.
  4. The figure below shows a graph in the x-y plane. Imagine it rotated around the $x$-axis to make a shell.
  5. The figure below repeats the graph shown in Exercise 3. Imagine it rotated about the $y$-axis to make a shell.
  6. The figure below shows two graphs. Foxus on the region contained between them and the $y$-axis. Imagine both graphs rotated about the $y$-axis to make a shell.

More About Cylinders

Ways Cylinders Can Vary

Though the size of a cylinder can vary in many ways, we will focus on their height and radius. Figure 8.3.? shows the same initial cylinder varying in these two ways: The left animation shows the cylinder's height varying while its base remains constant. The right animation shows the cylinder's radius varying while its height remains constant.


Figure 8.3.?. Two ways that a cylinder's size can vary: In height (left) and in radius (right).

Rate of change of cylinder's volume with respect to its varying height

Blah blah blah

Rate of change of cylinder's volume with respect to its varying radius

Blah Blah Blah

Quantifying Volumes of Solids

The animations showing different solids being filled illustrated the idea that approximate volume accumulates at a constant rate as $dx$ or $dy$ varies through intervals of length $\Delta x$ or $\Delta y$.

The exact accumulation of volume with respect to x or y will then be $V(a,x)=\int_a^x r_v(x)dx$ or $V(a,y)=\int_a^y r_v(y)dy$.

Our principle problem in computing the accumulation of volume is therefore to determine the rate of change of volume with respect to an independent variable for specific ways to fill a shell.

Rate of change of solid's volume with respect to cylinders' varying heights

Blah Blah Blah

Rate of change of solid's volume with respect to cylinders' varying radii

Blah Blah Blah

Accumulation of solid's volume with respect to its independent variable

Blah Blah Blah

Deciding on a Quantification Method

The way you envision filling a solid's shell will determine the mathematics you do to quantify the solid's volume. Therefore, you must consider the difficulty of describing a cylinder's base in relation to the independent variable your method dictates.

For some solids, the way you fill its shell will not make a difference in the difficulty of describing it mathematically. With other solids, one way of filling the solid's shell can lead to much simpler mathematics than other ways. We share two examples to illustrate this point.

Use example of $f(x)=\sin(x)$ rotated about the $x$-axis. We showed an approximation to shell's volume using cylinders whose heights vary. Show how using cylinders whose radii vary leads to using $x$ to locate one end of a cylinder and $\mathrm{asin}(\pi-x)$ to locate the other end of of the cylinder.

Exercise Set 8.3.2 (To Be Revised; remove reliance on B&C)

    The following exercises are excerpts from Briggs and Cochran (2nd Edition). Early exercises have you practice the "shell" method. Later exercises are drawn from a variety of sources.

    The purpose for this variety is so that you must choose a method that is consistent with how you envision the solid is generated and how you envision a bit of volume being generated.

    NOTE: We will value work that makes efficient use of function notation over work that repeatedly re-uses formulas and expressions.

  1. (Include a diagram!) Explain the difference between a shell created by what B&G call "the shell method" and a cylinder created by what B&G call the "slices" or "disks" method?

  2. Do the following for each of the (a)-(e) in which a region R is rotated about an axis in the indicated direction. Click here for checks on your answers to (a)-(d).

  3. a.

    b.
    c.
    d.
    e. (See solution to 41 after working on it)
    f.

  4. Each figure below indicates a solid of revolution. For each solid, use two different methods to define accumulating volume as a function of an independent variable ($x$ or $y$).
  5. (Show figures here)

  6. Examine this solution to the example problem discussed in class.

    1. Print this graph. Let $a=1, b=5, n=4$. On the printout of the graph of $g$, sketch the approximating segments to the graph between $x=a$ and $x=b$.
    2. Write out the summation in detail. Continue to use function notation!

    3. Explain how this sum approximates the arc length of $g$'s graph over the interval from $x=a$ to $x=b$.

    4. Explain how the line $r_g(x)=\dfrac{d}{dx}g(x)$ defines the rate of change function for $g$.

    5. Explain how the definition of $r_s$ works to give the rate of change of arc length of the graph of $g$ with respect to x as x varies from $x=a$ to $x=b$.

    6. Enter $s(-2,2)$. Why does GC give the answer it does?

    Use GC to do this for each of exercises 5 through 8:

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

  8. Function g defined as $g(x)=\ln(x)$. Interval:  $x=1 \text{ to }x=4$
  9. Function j defined as $j(x)=3\cos(10\sin(4x))$. Interval  $x=-4 \text{ to }x=4$.

  10. Function k defined as $k(y)=\dfrac{x^2}{1+x^3}$. Interval: $x=0 \text{ to }x=5$.
  11. Each function below is defined over an interval I. Assume that all values of x and y are a number of meters.
    Consider the surface that is generated by revolving the function's graph over that interval about the indicated axis.  Do the following for each function.

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

    2. $g(y)=3-y^2$ rotated about the y-axis, $0\le y \le 3$. (Enter $x=g(y)$ to graph x as a function of y.) What is $S(2.1)$? What does this number mean?
    3. $h(x)=1+2x$ rotated about the y-axis, $1\le y \le2$. (Graph $y=h(x)$; rotate that graph about the y-axis.) What is $S(1.5)$? What does this number mean?

    4. $j(y)=4-y^2$ rotated about the x-axis, $0\le y \le 2$. (Graph $x=j(y)$; rotate that graph about the x-axis. Consider the surface in two parts.)
  12. From B&C (6.6.35)
    In the design of solid objects (both artificial and natural), the object's surface area to volume (SAV) ratio is important. 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 tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) 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. Let the function f be defined as $f(x)=\sin(x)+0.3\cos(30x)$. What is the SAV ratio of the solid created by revolving the region bounded by the graph of f from $x=1$ to $x=2$ about the x-axis? Include the ends of the solid when computing its surface area.

Surface Area