< Previous Section Home Next Section >

# Section 8.3 Volumes of Regions in Space

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 variation in functions' values, that launched the calculus.

Newton thought of completed volumes and surface areas as having accumulated as they covaried with another quantity. Our treatment of volumes and surface areas of solids will build from Newton's approach.

To develop methods for quantifying the volume of regions in space, we will use ideas from Chapter 5 that we employed in Section 8.2 to find areas of regions in the plane.

We will define functions that model the accumulation of volume or surface area by determining the rate at which volume or surface area varies through moments of its independent variable.

In this section you will learn to conceptulize a solid's volume by a two-step process:
1. Think of the solid as having an empty shell that bounds a region in space
2. Fill the shell in a way that supports quantifying the filled region's volume.

## Different Ways to Make a Solid's Shell

It is natural to look at a solid and to think that nothing about it varies--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 quantifying its volume.

### Create a shell by varying an outline

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 cylinder's base and top are congruent and lie in parallel planes. The shape of the base (and top) do not matter. It only matters that they are congruent and lie in parallel planes.
• Every cross section taken parallel to the base is congruent to the base.

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.

Cylinders can be upright or slanted. We will employ only upright, or "right" cylinders to approximate a region bounded by a shell.

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

In this chapter, we shall mean "right cylinder" when we speak of cylinders.

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 constant 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 constant width and varying height.
Completed cylinders are shown in magenta. Varying cylinders are shown in blue.

We use "fill" in the sense of "fill a jar with marbles". The cylinders leave empty gaps within the shell. So the volume of our fills approximates the volume of the solid within the shell.

A much smaller value of $\Delta h$ produces smaller gaps and therefore produces a more accurate approximation 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 constant 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 constant radius and varying height.
Completed cylinders are shown in magenta. Varying cylinders are shown in blue.

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 constant 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 constant radius and varying height.
Completed cylinders are shown in magenta. Varying cylinders are shown in blue.

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 constant 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 constant height and varying radius.
Completed cylinders are shown in magenta. Varying cylinders are shown in blue.

Reflection 8.3.4. Figures 8.3.7 and 8.3.9 show the same shell being filled 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 $\Delta h$.

## 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.
1. Print two copies of the leaf.
2. Sketch one or more semi-circles that fit the description given above. See this hint if you are unsuccessful.
3. On the second copy of the leaf, sketch the shell that would be created by varying your semi-circle along the x-axis. Check yourself by viewing this figure.
4. What is the accumulating volume's independent variable?

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.
1. Print two copies of the leaf.
2. Sketch one or more rectangles that fit the description given above. See this hint if you are unsuccessful.
3. On the second copy of the leaf, sketch the shell that would be created by varying your rectangle along the x-axis. Check yourself by viewing this figure.
4. What is the accumulating volume's independent variable?

3. The figure below shows a graph in the x-y plane. Imagine it rotated around the y-axis to make a shell.
1. Print three copies of the graph. Mark $\Delta x$-intervals along the x-axis on the second and third printouts.
2. On the first printout, sketch the shell that will be made by rotating the graph as described.
3. On the second printout, sketch one cylinder over a $\Delta x$-interval so that:

• The cylinder's inner radius is the value of x at the beginning of the $\Delta x$-interval
• The cylinder's outer radius is the value of x as x varies by $dx$ through the $\Delta x$-interval.
• The cylinder's height throughout the $\Delta x$-interval is the value of $f(x)$ at the beginning of the interval
See this hint if you are unsuccessful. See this hint if you are still unsuccessful.
4. Notice that f's independent variable is x while the accumulating volume's independent variable also is x. Could you have used y as the accumulating volume's independent variable? Explain.

5. On the third printout, sketch the solid that would be created were cylinders to vary through very small $\Delta x$-intervals. Check yourself by viewing this figure.
4. The figure below shows a graph in the x-y plane. Imagine it rotated around the x-axis to make a shell.
1. Print three copies of the graph. Mark $\Delta x$-intervals along the x-axis on the second and third printouts.
2. On the first printout, sketch the shell that will be made by rotating the graph as described.
3. On the second printout, sketch one cylinder over a $\Delta x$-interval so that:

• The cylinder's radius throughout a $\Delta x$-interval is the value of $f(x)$ at the beginning of the interval
• The cylinder's height is $dx$ as x varies through a $\Delta x$-interval.
See this hint if you are unsuccessful. See this hint if you are still unsuccessful.
4. What is f's independent variable? What is the accumulating volume's independent variable? Explain why you could or could not have used the other variable as the accumulating volume's independent variable?
5. On the third printout, sketch the solid that would be created were cylinders to vary through very small $\Delta x$-intervals. Check yourself by viewing this figure.
5. The figure below repeats the graph shown in Exercise 3. Imagine it rotated around the y-axis to make a shell.
1. Print three copies of the graph. Mark $\Delta y$-intervals along the y-axis on the second and third printouts.
2. On the first printout, sketch the shell that will be made by rotating the graph as described.
3. On the second printout, sketch one cylinder over a $\Delta y$-interval so that:

• Its base is at the beginning of the $\Delta y$-interval
• Its radius throughout the $\Delta y$-interval is the value of x at which the base intersects the graph
• Its height is $dy$ as the value of y varies through the interval.
See this hint if you are unsuccessful. See this hint if you are still unsuccessful.
4. What is f's independent variable? What is the accumulating volume's independent variable? Explain why you could or could not have used the other variable as the accumulating volume's independent variable?
5. On the third printout, sketch the solid that would be created were cylinders to vary through very small $\Delta y$-intervals. Check yourself by viewing this figure.
6. Compare your small-$\Delta x$ sketch for Exercise 3 with your small-$\Delta y$ sketch in this exercise. Do these two methods give essentially the same volume for infinitesimally small values of $\Delta y$?
6. The figure below shows two graphs. Focus on the boundary of the region contained between them and the y-axis. Imagine both graphs rotated around the y-axis to make a shell.
1. Print three copies of the graph. Mark $\Delta y$-intervals along the y-axis on the second and third printouts.
2. On the first printout, sketch the shell that will be made by rotating the graph as described.
3. On the second printout, sketch one cylinder over a $\Delta y$-interval so that:

• The cylinder's outer radius begins on the graph of g at the beginning of a $\Delta y$-interval
• The cylinder's inner radius begins at the y-axis or the graph of f, whichever is appropriate.
• The cylinder's height is $dy$ as y varies through each $\Delta y$-interval.
See this hint if you are unsuccessful. See this hint if you are still unsuccessful.
4. What is the independent variable of f and g? What is the accumulating volume's independent variable? Explain why you could or could not have used the other variable as the accumulating volume's independent variable?
5. On the third printout, sketch the solid that would be created were cylinders to vary through very small $\Delta x$-intervals. Check yourself by viewing this figure.
7. The figure below repreats the one from Exercise 6. Imagine the graphs being rotated around the x-axis.
1. Print two copies of the figure.
2. On one printout, sketch the cylinders that would be created by letting y vary through intervals of size $\Delta y$. What is the accumulating volume's independent variable? Why?
3. On the second printout, sketch the cylinders that would be created by letting x vary through intervals of size $\Delta x$. What is the accumulating volume's independent variable? Why?

## 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 8.3.10. Two ways that a cylinder's size can vary:
• Its height varies while its base remains constant (left), or
• Its radius varies while its height remains constant (right).

The two cylinders start in identical states, with height $h_0$, inner radius $r_0$ and outer radius $r_1$, where $r_1\gt r_0\ge 0$. The left cylinder's height in Figure 8.3.10 increased by 1. The right cylinder's outer radius increased by 1. Let's examine their variations in volume.

The initial volume of both cylinders is area of base times height. So \begin{align} V_\text{init}&=\left(\pi r_1^2-\pi r_0^2\right)h_0\\[1ex] &=\pi h_0 \left(r_1^2-r_0^2\right).\end{align} The left cylinder's final volume after its height increased by 1 is \begin{align} V_L&=\pi \left(h_0+1\right) \left(r_1^2-r_0^2\right)\\[1ex] &=\pi h_0\left(r_1^2-r_0^2\right)+\pi\left(r_1^2-r_0^2\right)\\[1ex] &=V_\text{init}+\pi\left(r_1^2-r_0^2\right). \end{align}

The right cylinder's final volume after its radius increased by 1 is \begin{align} V_R&=\pi h_0 \left(\left(r_1+1\right)^2-r_0^2\right)\\[1ex] &=\pi h_0\left(r_1^2-r_0^2+2r_1+1\right)\\[1ex] &=\pi h_0\left(r_1^2-r_0^2\right)+\pi h_0\left(2r_1+1\right)\\[1ex]&=V_\text{init}+\pi h_0\left(2r_1+1\right).\end{align}

The cylinders' were identical at first. Their volumes began with the same value. Their volumes increased by different amounts with a change of 1 in height in the left cylinder and a change of 1 in radius in the right cylinder.

Therefore, a cylinder's rate of change of volume with respect to height and its rate of change of volume with respect to outer radius are different.

### Rate of change of cylinder's volume with respect to height when its height varies

The formula for volume of a cylinder is $$V=A_bh$$where $A_b$ is the area of the cylinder's base and h is the cylinder's height.

If the cylinder's base remains constant while its height varies, then volume is a function of height, so \begin{align}V(h)&=A_bh,\text{ and therefore}\\[1ex]r_v(h)&=A_b\end{align}

This says that when a cylinder varies so that its base remains constant while its height varies, the cylinder's rate of change of volume with respect to its height has the same numerical value as the area of its base.

We emphasize "... same numerical value ..." because the cylinder's rate of change of volume with respect to height is measured in cubic units per unit while the area of the base is measured in square units. They are different quantities having the same numerical value.

To repeat, a cylinder's rate of change of volume with respect to height is $r_v(h)=A_b$. The cylinder's change in volume over any $\Delta h$-interval is therefore $A_b dh$ as h varies by $dh$ through the interval.

When the base area of a circular cylinder is constant and the cylinder's height varies, the cylinder's rate of change of volume with respect to its height is constant.

The cylinder's change in volume over any $\Delta h$-interval is $r_v(h)dh$ as h varies by $dh$ through the interval. Accumulated volume within the solid's shell as h varies from $a$ to x is therefore $V(a,x)=\int_a^x r_v(h)dh$. (See a repeat of Figure 8.3.7 here.)

Notice that we defined V with two arguments. We included the initial value of accumulation as an argument in V's definition so that we do not need to define the intial value separately, with a line $a=a_0$. The accumulation's initial value is now part of the definition of V.

Instead of writing $$a=2, V(x)=\int_a^x r_v(t)dt$$ we defined V as $$V(a,x)=\int_a^x r_v(t)dt.$$ We can then write $V(2,x)$, $V(3,x)$, etc., without having to define different starting values for different evaluations of V.

We can also make statements like $V(3,7)=V(3,5)+V(5,7)$. If we feel adventurous, we could create a 3-d surface by graphing $z=V(x,y)$ in GC.

Reflection 8.3.5. We determined that the rate of change of a cylinder's volume with respect to its height is constant, and has a numerical value that is equal to the area of its base. Examine the animation in Figure 8.3.11. Does this cylinder's volume vary at a constant rate with respect to its height?

Figure 8.3.11. Does this cylinder's volume vary at a constant rate with respect to its height?

### Rate of change of cylinder's volume with respect to radius when its radius varies

Consider the cylinder below. The cylinder's height h and inner radius $r_0$ remain constant while its outer radius r varies.

The cylinder's volume as a function of outer radius is $$V(r)=\pi h\left(r^2-r_0^2\right)$$ where h and $r_0$ are constants.

The rate of change of the cylinder's volume with respect to its varying radius is therefore \begin{align}r_v(r)&=\pi h 2r\\[1ex] &=(2\pi r )h\\[1ex]&=\text{perimeter}\cdot \text{height}\end{align}

The value of $r_v(r)$, the cylinder's rate of change of volume with respect to its outer radius r, is $2\pi r h$.

Notice: $2\pi r h$ is the numerical value of the cylinder's outer surface area at each value of r as r varies. The cylinder's volume does not vary at a constant rate as its outer radius varies.

However, for sufficiently small values of $\Delta r$ the cylinder's rate of change of volume with respect to r is essentially constant. When $\Delta r$ has an infinitesimal value, the change in volume is essentially $r_v(r)dr$ as r varies by $dr$.

When a circular cylinder's height h is constant and the its outer radius r varies, the cylinder's rate of change of volume with respect to its outer radius is $r_v(r)=2\pi hr$.

The cylinder's variation in volume as $dr$ varies over sufficiently small $\Delta r$-intervals is essentially $r_v(r)dr$. Total accumulated volume within the solid's shell as r varies from $a$ to x is therefore $V(a,x)=\int_a^x r_v(r)dr$. (See a repeat of Figure 8.3.9 here.)

We emphasize again the phrase "...same numerical value". Rate of change of volume with respect to outer radius is measured in $\text{unit}^3/\text{unit}$. Surface area of a cylinder's outer side is measured in $\text{unit}^2$.

A cylinder's rate of change of volume with respect to radius and the cylinder's surface area as a function of radius are different quantities that have the same numerical value.

Finally, we state without demonstration two generalizations of the above development.

• A cylinder's rate of change of volume with respect to its height has the same numerical value as the area of its base, regardless of the base's shape.
• A cylinder's rate of change of volume with respect to its outer radius, when the enlarged base is similar to the original, has the same numerical value as the cylinder's outer surface area, regardless of the base's shape.

## Quantifying Volumes of Solids

The animations showing different solids being filled within a Cartesian coordinate system illustrate the idea that approximate volume accumulates at an essentially constant rate as $dx$ or $dy$ varies through sufficiently small intervals of the accumulating volumes' independent variable.

Remember that an accumulating volume's independent variable is determined by the method you choose to fill the solid's shell. It need not be the same as the independent variable of the function or functions whose graphs generate the shell.

If you vary accumulated volume by varying the value of x, then x is the accumulation's independent variable. If you vary accumulated volume by varying the value of y, then y is the accumulation's independent variable.

To quantify an accumulating volume we need mathematical descriptions of the functions whose graphs generate the solid's shell. Once we have actual functions we can use an appropriate method to represent, and hence calculate, heights, widths, and radii of cylinders for the method we choose.

### Example 1

Figure 8.3.12 repeats Figure 8.3.6. The axes in Figure 8.3.6 were not named, so we can name them in any convenient way. Take the left downward axis to hold values of positive x and the right downward axis to hold values of positive y.

Figure 8.3.12. Quarter circle in the x-y plane with radius $r=1$.
Take the left downward axis to be positive x and the right downward axis to be positive y.
Completed cylinders are shown in magenta. Varying cylinders are shown in blue
.

The quarter circle in Figure 8.3.12 has radius $r=1$. Its equation is $x^2+y^2=1,\,0\le x \le 1,\, 0\le y \le 1$. We can rewrite the equation to express y as a function of x, so that $y=f(x)$ where $f(x)=\sqrt{1-x^2},\, 0 \le x \le 1$. The quarter circle now is the graph of $y=f(x)$.

As the solid's accumulating volume varies, a square cylinder in blue is the only one that is varying in volume. It becomes magenta when it is "complete", after x (and hence $dx$) has varied through that $\Delta x$-interval.

A cylinder has a constant base area as x varies within the cylinder's $\Delta x$-interval. The base is a square with sides of length $f(x_b)$, where $x_b$ is the value of x at the beginning of the cylinder's $\Delta x$-interval. The cylinder's base area is $\left(f(x_b)\right)^2$.

The rate of change of volume with respect to height for a square cylinder is therefore $r_v(x)=\left(f(x)\right)^2$ for each value of x within a $\Delta x$-interval.

For sufficiently small values of $\Delta x$, the accumulated volume within a cylinder is $r_v(x)dx$ as the cylinder's height $dx$ varies. The total accumulation of volume from $a$ to x will therefore be $V(a,x)=\int_a^x r_v(t)dt$, or \begin{align}V(a,x)&=\int_a^x r_v(t)dt\\[1ex] &=\int_a^x \left(f(t)\right)^2dt\\[1ex] &=\int_a^x \left(\sqrt{1-t^2}\right)^2dt\\[1ex] &=\int_a^x\left(1-t^2\right)dt\end{align}

Because we defined $r_v(x)=\left(f(x)\right)^2$ and defined f as $f(x)=\sqrt{1-x^2}$, the first line of the above equation would have been sufficient to define $V(a,x)$. The remaining lines simply substitute the definitions of $r_v$ and f.

The left side of Figure 8.3.13 shows volume accumulating over infintesimal $\Delta x$-intervals. The right side shows the graph of accumulating volume as the value of x varies from 0 to 1.

Figure 8.3.13. (Left) The solid's volume accumulates smoothly as x varies from 0 to 1.
(Right) the graph of $V(0,x)$ as x varies from 0 to 1.

Reflection 8.3.6. Enter the definitions of f, $r_v$, and V into GC.
• Enter $y=V(0,x),\, 0\le x\le 1$. You should see the full graph as shown in Figure 8.3.13.
• What about the varying accumulation makes it sensible that the graph has the shape that it does?
• Enter $V(0,0.5)$. What does this number mean?
• Enter $V(0.5,1)$. What does this number mean?

### Example 2

Figure 8.3.14 repeats the graphs from Exercise 6 of Exercise Set 8.3.1. In this figure, $\,f(x)=1+\sin(x/2.5)$ and $g(x)=x^2/4$. The animation shows the cylinders that will fill the shell that these graphs make when rotated around the y-axis.

While varying, the cylinders have a constant height and a varying outer radius. The cylinder's height throughout each $\Delta x$-interval is the value of $f(x)-g(x)$ at the beginning of the interval. The outer radius is the value of x as x varies by $dx$ through the $\Delta x$-interval.

The volume of each cylinder (and thus of the accumulating volume) for infitesimal values of $\Delta x$ varies at a rate that is equal to the cylinder's outer surface area, which is $perimeter\cdot height$, or $(2\pi x)\left(f(x)-g(x)\right)$.

Figure 8.3.14. One cylinder varying in radius as x varies by $dx$ through each $\Delta x$-interval.
The graphs' point of intersection, determined graphically in GC, is $(3.4424,2.9625)$ to four decimal places.

Reflection 8.3.7. Explain why the height of each cylinder at the beginning of its $\Delta x$-interval is $f(x)-g(x)$.

Figure 8.3.15 adds two details to the discussion of Figure 8.3.14. First, it shows that the accumulating volume varies by $r_v(x)dx$ as x varies by $dx$ through $\Delta x$-intervals of infinitesimal length. Second, it shows that the function that evaluates accumulated volume from 0 to x is $V(0,x)=\int_0^x r_v(t)dt$. The graph on the right is of $y=V(0,x)$ as the value of x varies.

Figure 8.3.15.

Reflection 8.3.8. As noted in Figure 8.3.14, the x-coordinate of the graphs' point of intersection's is 3.4424. Use this fact, and the definition of $V$, to determine the total accumulated volume as x varies from 0 to 3.4424.

### Example 3

Figure 8.3.16 repeats the graphs from Figure 8.3.14 with f and g defined as $f(x)=1+2\sin(x/2.5)$ and $g(x)=x^2/4$. We again rotate the graphs around the y-axis. This time, however, we will fill the solid's shell with cylinders by letting y be the accumulating volume's independent variable.

Figure 8.3.16. The cylinder's height is $dy$ as y varies through a $\Delta y$-interval. The cylinder's width is the distance between the graphs at the beginning of the $\Delta y$-interval.

The cylinder's height is $dy$ as y varies through a $\Delta y$-interval. The cylinder's inner radius and outer radius are values of x. It requires an explanation to understand this clearly. Refer to Figure 8.3.17 for the following discussion.

Figure 8.3.17. Two views of a the cylinder that varies in Figure 8.3.16: Its appearance in 3 dimeinsions (right) and
the cylinder's relationships with graphs of $y=f(x)$ and $y=g(x)$.

Figure 8.3.17 shows two views of the cylinder that varies in Figure 8.3.16. The left side shows that the cylinder's height is $dy$ as y varies through a $\Delta y$-interval that begins at $y=y_0$.

It also shows that the inner circle has radius $f^{-1}(y_0)$ for the value of y at the beginning of a $\Delta y$-interval and its outer radius is $g^{-1}(y_0)$. The right side shows that the area of the cylinder's base is the area of circle bounded by the outside circle, including the hole, minus the area of the hole bounded by the inside circle. Why are we concerned with the area of the base? Because the area of the base is the rate of change of volume with respect to height of a cylinder whose base is constant while its height varies.

The cylinders in Figure 8.3.16 all have a constant base and varying height. The accumulating volume's rate of change is therefore the cylinder's base area. However, to determine the base area we must define inverse functions for f and g.

We must determine the function h so that $h(y)=f^{-1}(y)$ for $f(x)=1+2\sin(x/2.5),\, x\ge 0\text{ and }y\ge 1$. We do this by solving for x in terms of y: \begin{align} y&=1+2\sin\left(\frac{x}{2.5}\right)\\[1ex] y-1&=2\sin\left(\frac{x}{2.5}\right)\\[1ex] \frac{y-1}{2}&=\sin\left(\frac{x}{2.5}\right)\\[1ex] \mathrm{asin}\left(\frac{y-1}{2}\right)&=x/2.5\\[1ex] 2.5\mathrm{asin}\left(\frac{y-1}{2}\right)&=x\\[1ex] h(y)&=2.5\mathrm{asin}\left(\frac{y-1}{2}\right) \end{align}

For a given value of $y\ge 1$ in the range of f on the y-axis, the value of $h(y)$ is the value of x on the x-axis such that $f(x)=y$.

We also need a function k so that $k(y)=g^{-1}(y)$ for $g(x)=x^2/4$, $x\ge 0$. This is easy: \begin{align}y&=\frac{x^2}{4}\\[1ex] 4y&=x^2\\[1ex] \sqrt{4y}&=x\\[1ex] k(y)&=\sqrt{4y}\end{align}

Reflection 8.3.9. Define f, g, h, and k, as given, above in GC.

• Graph $y=f(x)$ in one color and $x=h(y)$ in another color. Do the graphs coincide? Exactly? What must you do to make the graphs coincide exactly?
• Graph $y=g(x)$ in one color and $x=k(y)$ in another color. Do the graphs coincide? Exactly? What must you do to make the graphs coincide exactly?
• What does it mean about any functions q and s when the graphs of $y=q(x)$ and $x=s(y)$ coincide exactly?

Refer again to Figure 8.3.17. The area of the outer circle, including the hole, of any cylinder is $\pi k(x)^2$. The area of the inner circle, bounding the hole must be defined in two parts: for values of y such that $0\le y \le 1$ (there is no hole) and for values of $y\gt 1$ for which $x\gt 0$ (there is a hole). We'll define a new function j so that $j(y)=0$ if $0 \le y \le 1$, $\, j(y)= h(y)$ if $y\gt 1$. $$j(y)=\begin{cases} 0 & \text{if 0\le y \le 1}\\[1ex]h(y) & \text{if y\gt 1}\end{cases}$$

The rate of change of volume with respect to height for any cylinder is the area of the cylinder's base. We therefore define $r_v$ as $$r_v(y)=\pi \left(k(y)^2-j(y)^2\right).$$

For sufficiently small values of $\Delta y$, the volume of each cylinder as y varies by $dy$ along the y-axis is essentially $r_v(y)dy$. The exact volume accumulated between $a$ and y as y varies is therefore $$V(a,y)=\int_a^y r_v(t)dt$$

With regard to Figure 8.3.16, the volume that accumulates from 0 to y within the shell of the solid is therefore $V(0,y)$.

Reflection 8.3.10. As noted in Figure 8.3.14, the y-coordinate of the graphs' point of intersection's is 2.9625. Use this fact, and the definitions of $r_v$ and $V$, above, to determine the total accumulated volume as y varies from 0 to 2.9625. Compare this volume to the volume you calculated in Reflection 8.3.8. They should be equal. Are they? Why?

This simple statement, $V(a,y)=\int_a^y r_v(t)dt$, depends on having defined all the functions f, g, h, j, and k derived above.

For you to understand the definition of $V$ as $V(a,y)=\int_a^y r_v(t)dt$, you must understand each of f, g, h, j, and k and how they work together to define $V$ so that $V(a,y)$ gives exact volume accumulated between $a$ and y as y varies.

There is a lesson to learn from Examples 2 and 3: Choose your independent variable for accumulation wisely! If possible, avoid having to use inverse functions.

## Choosing a Quantification Method

There is no fixed set of rules to follow in quantifying a solid's volume. But there are helpful guidelines:

• Think about ways you might fill the solid's shell before you do any mathematics.
• Examine the function or functions that determine the shell. Consider the ease or difficulty of using the functions or their inverses to quantify a cylinder's base area and height.
• If one way of filling a solid's shell requires inverting a complicated function, then try a different method to fill the solid's shell.

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 or height in relation to the accumulating volume's independent variable your method dictates.

Always keep this in mind:

1. The accumulated volume as the accumulation's independent variable varies from $a$ to $u$ is $V(a,u)$, where $$V(a,u)=\int_a^u r_v(t)dt$$
• If by varying $du$ a cylinder's height varies and its base remains constant, then $r_v(u)$ is the area of the cylinder's base.
• If by varying $du$ a cylinder's radius varies and its height remains constant, then $r_v(u)$ is the cylinder's outer surface area.
2. Statement 1 is true for any solid made be revolving a graph around an axis, regardless of the axis of rotation or the accumulating volume's independent variable
3. Statement 1 is true for any solid whose bases for all values of the accumulation volume's independent variable are similar.

## Exercise Set 8.3.2

Use the guidelines stated above for choosing a method to quantify the volume of a solid. Then apply your method, explaining each step in your solution.

NOTE: We will value work that makes efficient use of function notation over work that repeatedly re-uses formulas and expressions. Actually, using function notation will clarify your reasoning as you develop your solution.

1. Define an accumulation function for each of (a)-(e) that gives the accumulating volume as the independent variable varies through its stated domain.

In developing your solution:

• Graph the given functions and print your graphs.
• Sketch the shell that the exercise describes.
• Sketch one or more cylinders that will fill the shell.
• Define functions in GC that give a cylinder's rate of change of volume with respect to accumulating volume's independent variable.
• Define the function in GC that will give accumulated volume over any interval of the accumulating volume's independent variable.
• Print GC's display of your volume function's graph.
• Answer any other questions in the exercise. Write your answers on the printout of your graph.

1. $f(x)=x-x^2,\, 0≤x≤1,y=0$ rotated around the y-axis.
2. $g(x)=1/(1+x^2),\, y=0,\, x=0,\, x=2$. Rotate the graphs around the y-axis.
3. $h(x)=\sqrt{x},\, y=0,\, x=4$. Rotate the graphs around the x-axis.
4. The solid formed when a hole of radius 2 is drilled symetrically through the center of a sphere of radius 6.
5. Do parts (e) and (f) twice: once using x as the accumulating volume's independent variable, then a second time using y as the accumulating volume's indpendent variable.

6. $x=y,\, x=y^3,\, 0\le y \le 1$. Rotate the graphs around the x-axis.
7. $y=x^2,\, y=4-x,\, x=0,\, 0\le x\le \text{the$x$-coordinate of their point of intersection.}$ Rotate the graphs around the y-axis.

2. Our treatment of volume presumed unsigned volume. Answer the following questions regarding signed volume.

The figure below shows graphs of $y=f(x)$ and $y=g(x),\, 0\le x\le 2$ where $f(x)=1+\cos(x)$ and $g(x)=x^3$. Print two copies of the graphs.

1. On the first printout, sketch the shell that would be generated by rotating the graphs around the y-axis.

2. On the second printout, mark $\Delta x$ intervals on the x-axis.

3. Sketch the cylinders that are formed by x having varied from 0 to 2 along the x-axis.

4. Define $r_v$ as $r_v(x)=2\pi x\left(f(x)-g(x)\right)$. Which part of the accumulated volume grew at a positive rate of change? Which part of the accumulated volume grew at a negative rate of change? Explain.

5. Define a function whose graph shows signed volume as x varies from 0 to 2. Graph it in GC.

6. Modify the definition of $r_v$ so that $\int_0^x r_v(t)dt$ produces unsigned accumulated volume from 0 to x as x varies from 0 to 2. Then define a function whose graph shows unsigned volume from 0 to x as x varies from 0 to 2. Graph it in GC.

 < Previous Section Home Next Section >