Pump Up the Volume: Using Calculus to Solve 3-D Problems - Intermediate Integration Topics - Calculus II For Dummies

Calculus II For Dummies, 2nd Edition (2012)

Part III. Intermediate Integration Topics

Chapter 10. Pump Up the Volume: Using Calculus to Solve 3-D Problems

In This Chapter

arrow Understanding the meat-slicer method for finding volume

arrow Using inverses to make a problem easier to solve

arrow Solving problems with solids of revolution and surfaces of revolution

arrow Finding the space between two surfaces

arrow Considering the shell method for finding volume

In Chapter 9, I show you a bunch of different ways to use integrals to find area. In this chapter, you add a dimension by discovering how to use integrals to find volumes and surface areas of solids.

First, I show you how to find the volume of a solid by using the meat-slicer method, which is really a 3-D extension of the basic integration tactic you already know from Chapter 1: slicing an area into an infinite number of pieces and adding them up.

As with a real meat slicer, this method works best when the blade is slicing vertically — that is, perpendicular to the x-axis. So I also show you how to use inverses to rotate some solids into the proper position.

After that, I show you how to solve two common types of problems that calculus teachers just love: finding the volume of a solid of revolution and finding the area of a surface of revolution.

With these techniques in your back pocket, you move on to more complex problems, where a solid is described as the space between two surfaces. These problems are the 3-D equivalent of finding an area between two curves, which I discuss in Chapter 9.

To finish up, I give you an additional way to find the volume of a solid: the shell method. Then, I provide some practical perspective on all the methods in the chapter so that you know when to use them.

Slicing Your Way to Success

Did you ever marvel at the way in which a meat slicer turns an entire salami into dozens of tasty little paper-thin circles? Even if you’re a vegetarian, calculus provides you with an animal-friendly alternative: the meat-slicer method for measuring the volume of solids.

The meat-slicer method works best with solids that have similar cross sections. (I discuss this further in the following section.) Here’s the plan:

1. Find an expression that represents the area of a random cross section of the solid in terms of x.

2. Use this expression to build a definite integral (in terms of dx) that represents the volume of the solid.

3. Evaluate this integral.

Don’t worry if these steps don’t make a whole lot of sense yet. In this section, I show you when and how to use the meat-slicer method to find volumes that would be difficult or impossible without calculus.

Finding the volume of a solid with congruent cross sections

Before I get into calculus, I want to provide a little bit of background on finding the volume of solids. Spending a few minutes thinking about how volume is measured without calculus pays off big-time when you step into the calculus arena. This is strictly no-brainer stuff — some basic, solid geometry that you probably know already. So just lie back and coast through this section.

One of the simplest solids to find the volume of is a prism. A prism is a solid that has all congruent cross sections in the shape of a polygon. That is, no matter how you slice a prism parallel to its base, its cross section is the same shape and area as the base itself.

The formula for the volume of a prism is simply the area of the base times the height:

V = Ab · h

So if you have a triangular prism with a height of 3 inches and a base area of 2 square inches, its volume is 6 cubic inches.

This formula also works for cylinders — which are sort of prisms with a circular base — and generally any solid that has congruent cross sections. For example, the odd-looking solid in Figure 10-1 fits the bill nicely. In this case, you’re given the information that the area of the base is 7 cm2 and the height is 4 cm, so the volume of this solid is 28 cm3.

Figure 10-1:Finding the volume of an odd-looking solid with a constant height.

9781118161708-fg1001.eps

Finding the volume of a solid with congruent cross sections is always simple as long as you know two things:

check The area of the base — that is, the area of any cross section

check The height of the solid

Finding the volume of a solid with similar cross sections

In the previous section, you didn’t have to use any calculus brain cells. But now, suppose that you want to find the volume of the scary-looking hyperbolic cooling tower on the left side of Figure 10-2.

What makes this problem out of the reach of the formula for prisms and cylinders? In this case, slicing parallel to the base always results in the same shape — a circle — but the area may differ. That is, the solid has similar cross sections rather than congruent ones.

You can estimate this volume by slicing the solid into numerous cylinders, finding the volume of each cylinder by using the formula for constant-height solids, and adding these separate volumes. Of course, making more slices improves your estimate. And, as you may already suspect, adding the limit of an infinite number of slices gives you the exact volume of the solid.

Figure 10-2:Estimating the volume of a hyperbolic cooling tower by slicing it into cylindrical sections.

9781118161708-fg1002.eps

Hmmm . . . this is beginning to sound like a job for calculus. In fact, what I hint at in this section is the meat-slicer method, which works well for measuring solids that have similar cross sections.

tip.epsWhen a problem asks you to find the volume of a solid, look at the picture of the solid and figure out how to slice it up so that all the cross sections are similar. This is a good first step in understanding the problem so that you can solve it.

technicalstuff.epsTo measure weird-shaped solids that don’t have similar cross sections, you need multivariable calculus, which is the subject of Calculus III. See Chapter 14 for an overview of this topic.

Measuring the volume of a pyramid

Suppose that you want to find the volume of a pyramid with a 6-x-6-unit square base and a height of 3 units. Geometry tells you that you can use the following formula:

9781118161708-eq10001.eps

This formula works just fine, but it doesn’t give you insight into how to solve similar problems; it works only for pyramids. The meat-slicer method, however, provides an approach to the problem that you can generalize to use for many other types of solids.

To start out, I skewer this pyramid on the x-axis of a graph, as shown in Figure 10-3. Notice that the vertex of the pyramid is at the origin, and the center of the base is at the point (3, 0).

Figure 10-3: A pyramid skewered on the x-axis of a graph.

9781118161708-fg1003.eps

To find the exact volume of the pyramid, here’s what you do:

1. Find an expression that represents the area of a random cross section of the pyramid in terms of x.

At x = 1, the cross section is 22 = 4. At x = 2, it’s 42 = 16. And at x = 3, it’s 62 = 36. So generally speaking, the area of the cross section is:

A = (2x)2 = 4x2

2. Use this expression to build a definite integral that represents the volume of the pyramid.

In this case, the limits of integration are 0 and 3, so:

9781118161708-eq10002.eps

3. Evaluate this integral:

9781118161708-eq10003.eps

This is the same answer provided by the formula for the pyramid. But this method can be applied to a far wider variety of solids.

Measuring the volume of a weird solid

After you know the basic meat-slicer technique, you can apply it to any solid with a cross section that’s a function of x. In some cases, these solids are harder to describe than they are to measure. For example, have a look at Figure 10-4.

The solid in Figure 10-4 consists of two exponential curves — one described by the equation y = ex and the other described by placing the same curve directly in front of the x-axis — joined by straight lines. The other sides of the solid are bounded planes slicing perpendicularly in a variety of directions.

Figure 10-4: A solid based on two exponential curves in space.

9781118161708-fg1004.eps

Notice that when you slice this solid perpendicular with the x-axis, its cross section is always an isosceles right triangle. This is an easy shape to measure, so the slicing method works nicely to measure the volume of this solid. Here are the steps:

1. Find an expression that represents the area of a random cross section of the solid.

The triangle on the y-axis has a height and base of 1 — that is, e0. And the triangle on the line x = 1 has a height and base of e1, which is e. In general, the height and base of any cross section triangle is ex.

So here’s how to use the formula for the area of a triangle to find the area of a cross section in terms of x:

9781118161708-eq10004.eps

2. Use this expression to build a definite integral that represents the volume of the solid.

Now that you know how to measure the area of a cross section, integrate to add all the cross sections from x = 0 to x = 1:

9781118161708-eq10005.eps

3. Evaluate this integral to find the volume.

9781118161708-eq10006.eps

Turning a Problem on Its Side

When using a real meat slicer, you need to find a way to turn whatever you’re slicing on its side so that it fits in the machine. The same is true for calculus problems.

For example, suppose that you want to measure the volume of the solid shown in Figure 10-5. The base of this solid (light gray) is bounded on its sides by the function y = x4 between the x-axis at the bottom and y = 2 across the top. The figure is 3 units high, such that the cross section when you slice parallel with the x-axis is a series of isosceles triangles, each with a height of 3 and a base that’s the width across the function y = x4.

Figure 10-5:Using inverses to get a problem ready for the meat-slicer method.

9781118161708-fg1005.eps

The good news is that this solid has cross sections that are all similar triangles, so the meat-slicer method will work. Unfortunately, as the problem currently stands, you’d have to make your slices perpendicular to the y-axis. But to use the meat-slicer method, you must make your slices perpendicular to the x-axis.

To solve the problem, you first need to flip the solid over to the x-axis, as shown on the right side of Figure 10-5. The easiest way to do this is to use the inverse of the function y = x4. To find the inverse, switch x and y in the equation and solve for y:

9781118161708-eq10007.eps

warning_bomb.epsNote that the resulting equation 9781118161708-eq10008.eps in this case isn’t a function of x because a single x-value can produce more than one y-value. However, you can use this equation in conjunction with the meat-slicer method to find the volume that you’re looking for.

1. Find an expression that represents the area of a random cross section of the solid.

The cross section is an isosceles triangle with a height of 3 and a base of 9781118161708-eq10009.eps, so use the formula for the area of a triangle:

9781118161708-eq10010.eps

2. Use this expression to build a definite integral that represents the volume of the solid.

9781118161708-eq10011.eps

3. Solve the integral.

9781118161708-eq10012.eps

Now evaluate this expression:

9781118161708-eq10013.eps

Two Revolutionary Problems

Calculus professors are always on the lookout for new ways to torture their students. Okay, that’s a slight exaggeration. Still, sometimes it’s hard to fathom exactly why a problem without much practical use makes the Calculus Hall of Fame.

In this section, I show you how to tackle two problems of dubious practical value (unless you consider the practicality passing Calculus II!). First, I show you how to find the volume of a solid of revolution, which is a solid created by spinning a function around an axis. The meat-slicer method, which I discuss in the previous section, also applies to problems of this kind.

Next, I show you how to find the area of a surface of revolution, a surface created by spinning a function around an axis. Fortunately, a formula exists for finding or solving this type of problem.

Solidifying your understanding of solids of revolution

A solid of revolution is created by taking a function, or part of a function, and spinning it around an axis — in most cases, either the x-axis or the y-axis.

For example, the left side of Figure 10-6 shows the function y = 2 sin x between x = 0 and 9781118161708-eq10014.eps

Figure 10-6: A solid of revolution of y = 2 sin x around the x-axis.

9781118161708-fg1006.eps

tip.epsEvery solid of revolution has circular cross sections perpendicular to the axis of revolution. When the axis of revolution is the x-axis (or any other line that’s parallel with the x-axis), you can use the meat-slicer method directly, as I show you earlier in this chapter.

However, when the axis of revolution is the y-axis (or any other line that’s parallel with the y-axis), you need to modify the problem as I show you in the earlier section “Turning a Problem on Its Side.”

To find the volume of this solid of revolution, use the meat-slicer method:

1. Find an expression that represents the area of a random cross section of the solid (in terms of x).

This cross section is a circle with a radius of 2 sin x:

9781118161708-eq10015.eps

2. Use this expression to build a definite integral (in terms of dx) that represents the volume of the solid.

This time, the limits of integration are from 0 to 9781118161708-eq10016.eps:

9781118161708-eq10017.eps

3. Evaluate this integral by using the half-angle formula for sines, as I show you in Chapter 7:

9781118161708-eq10018.eps

Now evaluate:

9781118161708-eq10019.eps

So the volume of this solid of revolution is approximately 9.8696 cubic units.

Later in this chapter, I give you more practice measuring the volume of solids of revolution.

Skimming the surface of revolution

The nice thing about finding the area of a surface of revolution is that there’s a formula you can use. Memorize it and you’re halfway done.

To find the area of a surface of revolution between a and b, use the following formula:

9781118161708-eq10020.eps

This formula looks long and complicated, but it makes more sense when you spend a minute thinking about it. The integral is made from two pieces:

check The arc-length formula, which measures the length along the surface (see Chapter 9)

check The formula for the circumference of a circle, which measures the length around the surface

So multiplying these two pieces together is similar to multiplying length and width to find the area of a rectangle. In effect, the formula allows you to measure surface area as an infinite number of little rectangles.

When you’re measuring the surface of revolution of a function f(x) around the x-axis, substitute r = f(x) into the formula I gave you:

9781118161708-eq10021.eps

For example, suppose that you want to find the surface of revolution that’s shown in Figure 10-7.

To solve this problem, first note that for 9781118161708-eq10022.eps. So set up the problem as follows:

9781118161708-eq10023.eps

Figure 10-7:Measuring the surface of revolution of y= x3 between x= 0 and x = 1.

9781118161708-fg1007.eps

To start off, simplify the problem a bit:

9781118161708-eq10024.eps

You can solve this problem by using the following variable substitution:

9781118161708-eq10025.eps

Now substitute u for 1 + 9x4 and 9781118161708-eq10025a.eps for x3 dx into the equation:

9781118161708-eq10025b.eps

Notice that I change the limits of integration: When x = 0, u = 1. And when x = 1, u = 10.

9781118161708-eq10026.eps

Now you can perform the integration:

9781118161708-eq10027.eps

Finally, evaluate the definite integral:

9781118161708-eq10028.eps

Finding the Space Between

In Chapter 9, I show you how to find the area between two curves by subtracting one integral from another. This same principle applies in three dimensions to find the volume of a solid that falls between two different surfaces of revolution.

The meat-slicer method, which I describe earlier in this chapter, is useful for many problems of this kind. The trick is to find a way to describe the donut-shaped area of a cross section as the difference between two integrals: one integral that describes the whole shape minus another that describes the hole.

For example, suppose that you want to find the volume of the solid shown in Figure 10-8.

Figure 10-8: A vase-shaped solid between two surfaces of revolution.

9781118161708-fg1008.eps

This solid looks something like a bowl turned on its side. The outer edge is the solid of revolution around the x-axis for the function 9781118161708-eq10029.eps. The inner edge is the solid of revolution around the x-axis for the function 9781118161708-eq10030.eps. Here’s how to solve this problem:

1. Find an expression that represents the area of a random cross section of the solid.

That is, find the area of a circle with a radius of 9781118161708-eq10031.eps and subtract the area of a circle with a radius of 9781118161708-eq10032.eps:

9781118161708-eq10033.eps

2. Use this expression to build a definite integral that represents the volume of the solid.

The limits of integration this time are 0 and 4:

9781118161708-eq10034.eps

3. Solve the integral:

9781118161708-eq10035.eps

Now evaluate this expression:

9781118161708-eq10036.eps

Here’s a problem that brings together everything you’ve worked with from the meat-slicer method: Find the volume of the solid shown in Figure 10-9. This solid falls between the surface of revolution y = ln x and the surface of revolution 9781118161708-eq10037.eps, bounded below by y = 0 and above by y = 1.

The cross section of this solid is shown on the right-hand side of Figure 10-9: a circle with a hole in the middle.

Figure 10-9:Another solid formed between two surfaces of revolution.

9781118161708-fg1009.eps

Notice, however, that this cross section is perpendicular to the y-axis. To use the meat-slicer method, the cross section must be perpendicular to the x-axis. Modify the problem using inverses, as I show you in “Turning a Problem on Its Side”:

9781118161708-eq10038.eps

The resulting problem is shown in Figure 10-10.

Figure 10-10:Use inverses to rotate the problem from Figure 10-9 so you can use the meat-slicer method.

9781118161708-fg1010.eps

Now you can use the meat-slicer method to solve the problem:

1. Find an expression that represents the area of a random cross section of the solid.

That is, find the area of a circle with a radius of ex and subtract the area of a circle with a radius of 9781118161708-eq10039.eps. This is just geometry, but I take it slowly so you can see all the steps. Remember that the area of a circle is πr2:

9781118161708-eq10040.eps

2. Use this expression to build a definite integral that represents the volume of the solid.

The limits of integration are 0 and 1:

9781118161708-eq10041.eps

3. Evaluate the integral:

9781118161708-eq10042.eps

So the volume of this solid is approximately 2.9218 cubic units.

Playing the Shell Game

The shell method is an alternative to the meat-slicer method, which I discuss earlier in this chapter. It allows you to measure the volume of a solid by measuring the volume of many concentric surfaces of the volume, called “shells.”

Although the shell method works only for solids with circular cross sections, it’s ideal for solids of revolution around the y-axis, because you don’t have to use inverses of functions, as I show you in “Turning a Problem on Its Side.” Here’s how it works:

1. Find an expression that represents the area of a random shell of the solid in terms of x.

2. Use this expression to build a definite integral (in terms of dx) that represents the volume of the solid.

3. Evaluate this integral.

As you can see, this method resembles the meat-slicer method. The main difference is that you’re measuring the area of shells instead of cross sections.

Peeling and measuring a can of soup

You can use a can of soup — or any other can that has a paper label on it — as a handy visual aid to give you insight into how the shell method works. To start out, go to the pantry and get a can of soup.

Suppose that your can of soup is industrial size, with a radius of 3 inches and a height of 8 inches. You can use the formula for a cylinder to figure out its volume as follows:

V = Ab · h = 32π · 8 = 72π

Another option is the meat-slicer method, as I show you earlier in this chapter. A third option, which I focus on here, is the shell method.

To understand the shell method, slice the can’s paper label vertically, and carefully remove it from the can, as shown in Figure 10-11. (While you’re at it, take a moment to read the label so that you’re not left with “mystery soup.”)

Notice that the label is simply a rectangle. Its shorter side is equal in length to the height of the can (8 inches) and its longer side is equal to the circumference (2π · 3 inches = 6π inches). So the area of this rectangle is 48π square inches.

Now here’s the crucial step: Imagine that the entire can is made up of infinitely many labels wrapped concentrically around each other, all the way to its core. The area of each of these rectangles is:

A = 2π x · 8 = 16π x

Figure 10-11:Removing the label from a can of soup can help you understand the shell method.

9781118161708-fg1011.eps

The variable x in this case is any possible radius, from 0 (the radius of the circle at the very center of the can) to 3 (the radius of the circle at the outer edge). Here’s how you use the shell method, step by step, to find the volume of the can:

1. Find an expression that represents the area of a random shell of the can (in terms of x):

A = 2π x · 8 = 16π x

2. Use this expression to build a definite integral (in terms of dx) that represents the volume of the can.

Remember that with the shell method, you’re adding up all the shells from the center (where the radius is 0) to the outer edge (where the radius is 3). So use these numbers as the limits of integration:

9781118161708-eq10043.eps

3. Evaluate this integral:

9781118161708-eq10044.eps

Now evaluate this expression:

= 8π (3)2 – 0 = 72π

The shell method verifies that the volume of the can is 72π cubic inches.

Using the shell method

One advantage of the shell method over the meat-slicer method comes into play when you’re measuring a volume of revolution around the y-axis.

Earlier in this chapter, I tell you that the meat-slicer method works best when a solid is on its side — that is, when you can slice it perpendicular to the x-axis. But when the similar cross sections of a solid are perpendicular to the y-axis, you need to use inverses to realign the problem before you can start slicing. (See the earlier section “Turning a Problem on Its Side” for more details.)

This realignment step isn’t necessary for the shell method. This makes the shell method ideal for measuring solids of revolution around the y-axis. For example, suppose that you want to measure the volume of the solid shown in Figure 10-12.

Figure 10-12:Using the shell method to find the volume of a solid of revolution.

9781118161708-fg1012.eps

Here’s how the shell method can give you a solution without using inverses:

1. Find an expression that represents the area of a random shell of the solid (in terms of x).

Remember that each shell is a rectangle with two different sides: One side is the height of the function at x — that is, cos x. The other is the circumference of the solid at x — that is, 2πx. So to find the area of a shell, multiply these two numbers together:

A = 2πx cos x

2. Use this expression to build a definite integral (in terms of dx) that represents the volume of the solid.

In this case, remember that you’re adding up all the shells from the center (at x = 0) to the outer edge (at 9781118161708-eq10045.eps).

9781118161708-eq10046.eps

3. Evaluate the integral.

This integral is pretty easy to solve using integration by parts:

9781118161708-eq10047.eps

Now evaluate this expression:

9781118161708-eq10048.eps

So the volume of the solid is approximately 0.5708 cubic units.

Knowing When and How to Solve 3-D Problems

Because students are so often confused when it comes to solving 3-D calculus problems, here’s a final perspective on all the methods in this chapter, and how to choose among them.

First, remember that every problem in this chapter falls into one of these two categories:

check Finding the area of a surface of revolution

check Finding a volume of a solid

In the first case, use the formula I provide earlier in this chapter, in “Skimming the surface of revolution.”

In the second case, remember that the key to measuring the volume of any solid is to slice it up in the direction where it has similar cross sections whose area can be measured easily — for example, a circle, a square, or a triangle. So your first question is whether these similar cross sections are arranged horizontally or vertically.

check Horizontally means that the solid is already in position for the meat-slicer method. (If it’s helpful, imagine slicing salami in a meat-slicer. The salami must be aligned lying on its side — that is, horizontally — before you can begin slicing.)

check Vertically means that the solid is standing upright so that the slices are stacked on top of each other.

When the cross sections are arranged horizontally, the meat-slicer method is the easiest way to handle the problem (see “Slicing Your Way to Success” earlier in this chapter).

When the cross sections are arranged vertically, however, your next question is whether these cross sections are circles:

check If the cross sections are not circles, you must use inverses to flip the solid in the horizontal direction (as I discuss in “Turning a Problem on Its Side”).

check If they are circles, you can either use inverses to flip the solid in the horizontal direction (as I discuss in “Turning a Problem on Its Side”) or use the shell method (as I discuss in “Playing the Shell Game”).