Tag Archives: volume

Is this going to be on the exam?

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confused-teacherRecently there was a discussion on the AP Calculus Community bulletin board regarding whether it was necessary or desirable to have students do curve sketching starting with the equation and ending with a graph with all the appropriate features – increasing/decreasing, concavity, extreme values etc., etc. – included. As this is kind of question that has not been asked on the AP Calculus exam, should the teacher have his students do problems like these?

The teacher correctly observed that while all the individual features of a graph are tested, students are rarely, if ever, expected to put it all together. He observed that making up such questions is difficult because getting “nice” numbers is difficult.

Replies ran from No, curve sketching should go the way of log and trig tables, to Yes, because it helps connect f. f ‘ and f ‘’, and to skip the messy ones and concentrate on the connections and why things work the way they do. Most people seemed to settle on that last idea; as I did. As for finding questions with “nice” numbers, look in other textbooks and steal borrow their examples.

But there is another consideration with this and other topics. Folks are always asking why such-and-such a topic is not tested on the AP Calculus exam and why not.

The AP Calculus program is not the arbiter of what students need to know about first-year calculus or what you may include in your course. That said, if you’re teaching an AP course you should do your best to have your students learn everything listed in the 2019 Course and Exam Description book and be aware of how those topics are tested – the style and format of the questions. This does not limit you in what else you may think important and want your students to know. You are free to include other topics as time permits.

Other considerations go into choosing items for the exams. A big consideration is writing questions that can be scored fairly.  Here are some thoughts on this by topic.

Curve Sketching

If a question consisted of just an equation and the directions that the student should draw a graph, how do you score it? How accurate does the graph need to be? Exactly what needs to be included?

An even bigger concern is what do you do if a student makes a small mistake, maybe just miscopies the equation? The problem may have become easier (say, an asymptote goes missing in the miscopied equation and if there is a point or two for dealing with asymptotes – what becomes of those points?) Is it fair to the student to lose points for something his small mistake made it unnecessary for him to consider? Or if the mistake makes the question so difficult it cannot be solved by hand, what happens then? Either way, the student knows what to do, yet cannot show that to the reader.

To overcome problems like these, the questions include several parts usually unrelated to each other, so that a mistake in one part does not make it impossible to earn any subsequent points. All the main ideas related to derivatives and graphing are tested somewhere on the exam, if not in the free-response section, then as a multiple-choice question.

(Where the parts are related, a wrong answer from one part, usually just a number, imported into the next part is considered correct for the second part and the reader then can determine if the student knows the concept and procedure for that part.)

Optimization

A big topic in derivative applications is optimization. Questions on optimization typically present a “real life” situation such as something must be built for the lowest cost or using the least material. The last question of this type was in 1982 (1982 AB 6, BC 3 same question). The question is 3.5 lines long and has no parts – just “find the cost of the least expensive tank.”

The problem here is the same as with curve sketching. The first thing the student must do is write the equation to be optimized. If the student does that incorrectly, there is no way to survive, and no way to grade the problem. While it is fair to not to award points for not writing the correct equation, it is not fair to deduct other points that the student could earn had he written the correct equation.

The main tool for optimizing is to find the extreme value of the function; that is tested on every exam. So here is a topic that you certainly may include the full question in you course, but the concepts will be tested in other ways on the exam.

The epsilon-delta definition of limit

I think the reason that this topic is not tested is slightly different. If the function for which you are trying to “prove” the limit is linear, then \displaystyle \delta =\frac{\varepsilon }{\left| m \right|} where m is the slope of the line – there is nothing to do beside memorize the formula. If the function is not linear, then the algebraic gymnastics necessary are too complicated and differ greatly depending on the function. You would be testing whether the student knew the appropriate “trick.”

Furthermore, in a multiple-choice question, the distractor that gives the smallest value of must be correct (even if a larger value is also correct).

Moreover, finding the epsilon-delta relationship is not what’s important about the definition of limit. Understanding how the existence of such a relationship say “gets closer to” or “approaches” in symbols and guarantees that the limit exists is important.

Volumes using the Shell Method

I have no idea why this topic is not included. It was before 1998. The only reason I can think of is that the method is so unlike anything else in calculus (except radial density), that it was eliminated for that reason.

This is a topic that students should know about. Consider showing it too them when you are doing volumes or after the exam. Their college teachers may like them to know it.

Integration by Parts on the AB exam

Integration by Parts is considered a second semester topic. Since AB is considered a one-semester course, Integration by Parts is tested on the BC exam, but not the AB exam. Even on the BC exam it is no longer covered in much depth: two- or more step integrals, the tabular method, and reduction formulas are not tested.

This is a topic that you can include in AB if you have time or after the exam or expand upon in a BC class.

Newton’s Method, Work, and other applications of integrals and derivatives

There are a great number of applications of integrals and derivatives. Some that were included on the exams previously are no longer listed. And that’s the answer right there: in fairness, you must tell students (and teachers) what applications to include and what will be tested. It is not fair to wing in some new application and expect nearly half a million students to be able to handle it.

Also, remember when looking through older exams, especially those from before 1998, that some of the topics are not on the current course description and will not be tested on the exams.

Solution of differential equations by methods other than separation of variables

Differential equations are a huge and important area of calculus. The beginning courses, AB and BC, try to give students a brief introduction to differential equations. The idea, I think, is like a survey course in English Literature or World History: there is no time to dig deeply, but the is an attempt to show the main parts of the subject.

While the choices are somewhat arbitrary, the College Board regularly consults with college and university mathematics departments about what to include and not include. The relatively minor changes in the new course description are evidence of this continuing collaboration. Any changes are usually announced two years in advance. (The recent addition of density problems unannounced, notwithstanding.) So, find a balance for yourself. Cover (or better yet, uncover) the ideas and concepts in the course description and if there if a topic you particularly like or think will help your students’ understanding of the calculus, by all means include it.

PS: Please scroll down and read Verge Cornelius’ great comment below.

Happy Holiday to everyone. There is no post scheduled for next week; I will resume in the new year. As always, I like to hear from you. If you have anything calculus-wise you would like me to write about, please let me know and I’ll see what I can come up with. You may email me at lnmcmullin@aol.com

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Area and Volume Questions

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AP Type Questions 4

Given equations that define a region in the plane students are asked to find its area and the volume of the solid formed when the region is revolved around a line or used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every year since 1969 on the AB exam and all but one year on the BC exam.

If this appears on the calculator active section, it is expected that the definite integrals will be evaluated on a calculator. Students should write the definite integral with limits on their paper and put its value after it.  It is not required to give the antiderivative and if students give an incorrect antiderivative they will lose credit even if the final answer is (somehow) correct.

There is a calculator program available that will give the set-up and not just the answer so recently this question has been on the no calculator allowed section. (The good news is that the integrals will be easy or they will be set-up but do not integrate questions.)

What students should be able to do:

  • Find the intersection(s) of the graphs and use them as limits of integration (calculator equation solving). Write the equation followed by the solution; showing work is not required. Usually no credit is earned until the solution is used in context (as a limit of integration).
  • Find the area of the region between the graph and the x-axis or between two graphs.
  • Find the volume when the region is revolved around a line, not necessarily an axis, by the disk/washer method. (Shell method is never necessary, but is eligible for full credit if properly used).
  • Find the volume of a solid with regular cross-sections whose base is the region between the curves. But see 2009 AB 4(b)
  • Find the equation of a vertical line that divides the region in half (area or volume). This involves setting up and solving an integral equation where the limit is the variable for which the equation is solved.
  • For BC only – find the area of a region bounded by polar curves:

\displaystyle A=\tfrac{1}{2}{{\int_{{{t}_{1}}}^{{{t}_{2}}}{\left( r\left( t \right) \right)}}^{2}}dt

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

For some previous posts on this subject see January 9, 11, 2013

Painting a Point

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Accumulation 7: An application (of paint)

Suppose you started with a point, the origin to be specific, and painted it. You put on layers and layers of paint until your point grows to a sphere with radius r. Let’s stop and admire your work part way through the job; at this point the radius is {{x}_{i}} and 0\le {{x}_{i}}\le r.

How much paint will you need for the next layer?

Easy: you need an amount equal to the surface area of the sphere, 4\pi {{x}_{i}}^{2}, times the thickness of the paint. As everyone knows paint is thin, specifically \Delta x thin. So we add an amount of paint to the sphere equal to 4\pi {{x}^{2}}\left( \Delta x \right).

The volume of the final sphere must be the same as the total amount of paint. The total amount of paint must be the (Riemann) sum of all the layers or \displaystyle \sum\limits_{i=1}^{n}{4\pi {{x}_{i}}^{2}\left( \Delta x \right)}. As usual \Delta x is very thin, tending to zero as a matter of fact, so the amount of paint must be

\displaystyle \underset{\Delta x\to 0}{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{4\pi {{x}_{i}}^{2}\left( \Delta x \right)=\int_{0}^{r}{4\pi {{x}^{2}}}dx=\left. \tfrac{4}{3}\pi {{x}^{3}} \right|_{0}^{r}=\tfrac{4}{3}\pi {{r}^{3}}}.

A standard related rate problem is to show that the rate of change of the volume of a sphere is proportional to its surface area – the constant of proportionality is dr/dt. So it should not be a surprise that the integral of this rate of change of the surface area is the volume. The integral of a rate of change is the amount of change.

Interestingly this approach works other places as long as you properly define “radius:”

  • A circle centered at the origin with radius x and perimeter of 2\pi x, gains area at a rate equal to its perimeter times the “thickness of the edge” \Delta x: \displaystyle A=\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{k=1}^{n}{2\pi x\Delta x}=\int_{0}^{r}{2\pi x\,dx=\pi {{r}^{2}}}
  • A square centered at the origin with “radius” x with sides whose length are s =2 x, gains area at a rate equal to its perimeter (8x) times the “thickness of the edge” \Delta x:  \displaystyle A=\underset{n\to \infty }{\mathop{\text{lim}}}\,\sum\limits_{k=1}^{n}{8x\Delta x=\int_{0}^{x}{8x\,dx}}=4{{x}^{2}}={{(2x)}^{2}}={{s}^{2}}
  • A cube centered at the origin with “radius” x and edges of length 2x, gains volume at a rate equal to its surface area, 6(4{{x}^{2}}), times the “thickness of the face” \Delta x – think paint again: V=\underset{n\to \infty }{\mathop{\text{lim}}}\,\sum\limits_{k=1}^{n}{\text{6(4}{{\text{x}}^{2}})\Delta x}=\int_{0}^{x}{24{{x}^{2}}dx}=8{{x}^{3}}={{(2x)}^{3}}={{s}^{3}}

Volumes of Revolution

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Applications of Integration 3 – Volumes of Rotation

In our last post we discussed volumes of figures with regular cross sections. Many common figures can be analyzed as some region rotated around a line, possibly one of its edges. For example, the region between the x-axis and the segment with equation y=\frac{r}{h}x from x = 0 to
x = h when rotated around the x-axis generates a cone with height h and a base with a radius of r. This has regular cross-sections which are circles for any value of x between 0 and h. Their areas are. So based on the idea in my previous post the volume is the integral of the cross-section area times the thickness:

\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{\pi {{\left( \tfrac{r}{h}{{x}_{i}} \right)}^{2}}\Delta {{x}_{i}}}=\int_{0}^{h}{\pi {{\left( \tfrac{r}{h}x \right)}^{2}}}dx=\tfrac{1}{3}\pi {{r}^{2}}h

Since circular cross-sections are very common (at least in calculus books) this is often treated as a separate topic with its own ideas and formulas. It really is not. It’s the basic idea applied to figures with circular sections; the cross-section area as a function of x is \pi {{\left( r(x) \right)}^{2}}; the thickness is \Delta x

\displaystyle V=\int_{a}^{b}{\pi {{\left( r\left( x \right) \right)}^{2}}dx}.

If the figure has a hole through it one subtracts the inside volume (radius = r\left( x \right) ) from the outside volume (radius = R\left( x \right)). This may be done with two integrals or combined into one.

\displaystyle \int_{a}^{b}{\pi {{\left( R\left( x \right) \right)}^{2}}dx}-\int_{a}^{b}{\pi {{\left( r\left( x \right) \right)}^{2}}}dx=\int_{a}^{b}{\pi {{\left( R\left( x \right) \right)}^{2}}-\pi {{\left( r\left( x \right) \right)}^{2}}dx}

Looking at the last integral we see that this is the same as treating the cross-section as an annulus (aka a “washer”). The \pi is often factored out in front of the integral sign to make things look neater; I suggest you leave it where it is to remind students that they are subtracting the areas of two circles.

To help students see what the figures look like you can do several things. You can cut the region out of light cardboard and tape it to the end of a pencil. Then rotate the pencil quickly to see the volume. Another way is to use a good graphing program, such as Winplot, which will let you see a three-dimensional figure being formed. One of my favorites is to use paper wedding bells that start flat and then open into a three-dimensional figure.

Volume of Solids with Regular Cross-sections

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Applications of Integration 2 – Solids with Regular Cross-sections

In Vino Veritas. And not only truth, but the start of an important application of the calculus. After Johannes Kepler’s (1571–1630) first wife died he decided to marry again. Naturally, he bought wine for the festivities. There was some disagreement with the wine merchant concerning how he calculated the volume of the barrel of wine. This led Kepler, a mathematician, to study the problem of finding the volume of barrels. The barrels, then as now, had circular cross sections of different diameters. His idea was to consider the wine as a stack of thin cylinders (with height of as we would say today). He then calculated the volume of each cylinder and added them together to find the volume of the barrel. There is a nice animation of the idea and more information on Kepler here.

The basic problem is this: you have a solid object whose volume you need to compute. If you can slice the object perpendicular to the axis in such a way that you get regular cross sections (i.e. similar figures) whose area you can compute, then by multiplying the area times the thickness and adding the results, you can have the volume. This kind of addition is done nicely by integration, since the volume of the slices form a Riemann sum.

You need to set up a coordinate system so that you can slice the figure into thin slices (whose width is ) and express the cross-section’s dimensions as a function of x and then the area as A(x). The result always looks like this where A(x) is the area of the cross-section:

\displaystyle \underset{n\to 0}{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{A\left( {{x}_{i}} \right)\Delta {{x}_{i}}}=\int_{a}^{b}{A\left( x \right)dx}

Think of this as the integral of the cross-section area times the thickness

Since you are probably not allowed to have wine in your calculus class, you may want to illustrate the idea some way other than using a wine barrel. Think of a loaf of sliced bread, or a deck of cards. Find some object that the students can measure and calculate the volume, such as a tree trunk, a statue, or a Coke-Cola bottle.

Another interesting exercise is to develop (prove) the formulas for volumes that students have known and used without proof since grade school: a cone, a pyramid, a sphere, the frustum of a cone.