Area and Volume Problems (Type 4)

AP Type Questions 4: Area and Volume

Given equations that define a region in the plane students are asked to find its area, the volume of the solid formed when the region is revolved around a line, and/or the region is used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every year since year one (1969) on the AB exam and almost every year on the BC exam. You can be fairly sure that if a free-response question on areas and volumes does not appear, the topic will be tested on the multiple-choice section.

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 (e.g., as a limit of integration). Students should know how to store and recall these values to save time and avoid copy errors.
  • 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 or an edge of the region, by the disk/washer method. See “Subtract the Hole from the Whole”
  • The cylindrical shell method will never be necessary for a question on the AP exams 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. For an interesting variation on this idea 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 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=\frac{1}{2}{{\int_{{{{\theta }_{1}}}}^{{{{\theta }_{2}}}}{{\left( {r\left( \theta \right)} \right)}}}^{2}}d\theta
  • For BC only – Find perimeter using arc length integral

If this question 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 a student gives 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 in this case the integrals will be easy, or they will be set-up-but-do-not-integrate questions.)

Occasionally, other type questions have been included as a part of this question. See 2016 AB5/BC5 which included an average value question and a related rate question along with finding the volume.

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 911, 2013 and “Subtract the Hole from the Whole” of December 6, 2016.


The Area and Volume question covers topics from Unit 6 of the CED .


Free-response questions:

  • Variations: 2009 AB 4, Don’t overlook this one, especially part (b)
  • 2016 AB5/BC5,
  • 2017 AB 1 (using a table),
  • 2018 AB 5 – average rate of change, L’Hospital’s Rule
  • 2019 AB 5
  • Perimeter parametric curves 2011 BC 3 and 2014 BC 5
  • Area in polar form 2017 BC 5, 2018 BC 5, 20129 BC 2
  • 2021 AB 4/ BC 4
  • 2022 AB2 – area, volume, inc/dec analysis, and related rate.

Multiple-choice questions from non-secure exams:

  • 2008 AB 83 (Use absolute value),
  • 2012 AB 10, 92
  • 2012 BC 87, 92 (Polar area)

Revised March 12, 2012, March 22, 2022


…but what does it look like?

It will soon be time to teach about finding the volumes of solid figures using integration techniques. Here is a list of links to posts that will help your students what these figures look like and how they are generated.

Visualizing Solid Figures 1 Here are ideas for making physical models of solid figures. These make good projects for students.

A Little Calculus is an iPad app that does an excellent job in helping students visualize many of the concepts of the calculus. Volumes with regular cross section, disk method, washer method, cylindrical shells are all illustrated.

The first illustrations show square cross sections on a semicircular base. The base is in the lower part and the solid in the upper. By using the plus and minus button (lower right) you can increase or decrease the number of sections in real time and see the figures change. The upper figure may be rotated by moving your finger on the screen.

The illustration below shows a washer situation.


The following older posts show how to use Winplot to generate and explore solid figures. Unfortunately, Winplot seems to have gone out of favor. I’m not sure why; it is one of the best. I still use it and like it. You may download Winplot here for free (PC only).

Visualizing Solid Figures 2 This post demonstrates how to use Winplot to generate solids with regular cross sections and solids of rotation.

Visualizing Solid Figures 3 The washer method is illustrated using Winplot. These post all relate to finding volumes by washers: Subtract the Hole from the Whole and Does Simplifying Make Things Simpler?

Visualizing Solid Figures 4 Using Winplot to see the method cylindrical shells. Note that this method is not tested on either the AB or BC Calculus exams, so you do not have to teach it. Many teachers present this topic after the exams are given. As a footnote you may also find Why You Never Need Cylindrical Shells interesting. (However, this is not the reason it is not tested on the AP Calculus exams.)

Visualizing Solid Figures 5 An exercise demonstrating how “half” can mean different things and shows that how the figures are generated makes a difference.

Adapting 2021 AB 3 / BC 3

Three of nine. Continuing the series started in the last two posts, this post looks at the AB Calculus 2021 exam question AB 3 / BC 3. The series considers each question with the aim of showing ways to use the question in with your class as is, or by adapting and expanding it. Like most of the AP Exam questions there is a lot more you can ask from the stem and a lot of other calculus you can discuss.

2021 AB 3 / BC 3

This question is an Area and Volume question (Type 4) and includes topics from Unit 8 of the current Course and Exam Description. Typically, students are given a region bounded by a curve and an line and asked to find its area and its volume when revolved around a line. But there is an added concept here that we will look at first.

The stem is:

First, let’s consider the c. This is a family of functions question. Family of function questions appear now and then. They are discussed in the post on Other Problems (Type 7) and topics from Unit 8 of the current Course and Exam Description. My favorite example is 1998 AB 2, BC 2. Also see Good Question 2 and its continuation.

If we consider the function with c = 1 to be the parent function \displaystyle P\left( x \right)=x\sqrt{{4-{{x}^{2}}}} then the other members of the family are all of the form \displaystyle c\cdot P\left( x \right). The c has the same effect as the amplitude of a sine or cosine function:

  • The x-axis intercepts are unchanged.
  • If |c| > 1, the graph is stretched away from the x-axis.
  • If 0 < |c| < 1, the graph is compressed towards the x-axis.
  • And if c < 0, the graph is reflected over the x-axis.

All of this should be familiar to the students from their work in trigonometry. This is a good place to review those ideas. Some suggestions on how to expand on this will be given below.

Part (a): Students were asked to find the area of the region enclosed by the graph and the x-axis for a particular value of c. Substitute that value and you have a straightforward area problem.

Discussion and ideas for adapting this question:

  • The integration requires a simple u-substitution: good practice.
  • You can change the value of c > 0 and find the resulting area.
  • You can change the value of c < 0 and find the resulting area. This uses the upper-curve-minus-the-lower-curve idea with the upper curve being the x-axis (y = 0).
  • Ask students to find a general expression for the area in terms of c and the area of P(x).
  • Another thing you can do is ask the students to find the vertical line that cuts the region in half. (Sometimes asked on exam questions).
  • Also, you could ask for the equation of the horizontal line that cuts the region in half. This is the average value of the function on the interval. See these post 1, 2, 3, and this activity 4.

Part (b): This question gave the derivative of y(x) and the radius of the largest cross-sectional circular slice. Students were asked for the corresponding value of c. This is really an extreme value problem. Setting the derivative equal to zero and solving the equation gives the x-value for the location of the maximum. Substituting this value into y(x) and putting this equal to the given maximum value, and you can solve for the value of c.  

(Calculating the derivative is not being tested here. The derivative is given so that a student who does not calculate the derivative correctly, can earn the points for this part. An incorrect derivative could make the rest much more difficult.)

Discussion and ideas for adapting this question:

  • This is a good problem for helping students plan their work, before they do it.
  • Changing the maximum value is another adaption. This may require calculator work; the numbers in the question were chosen carefully so that the computation could be done by hand. Nevertheless, doing so makes for good calculator practice.

Part (c): Students were asked for the value of c that produces a volume of 2π. This may be done by setting up the volume by disks integral in terms of c, integrating, setting the result equal to 2π, and solving for c.

Discussion and ideas for adapting this question:

  • Another place to practice planning the work.
  • The integration requires integrating a polynomial function. Not difficult, but along with the u-substitution in part (a), you have an example to show people that students still must do algebra and find antiderivatives.
  • Ask students to find a general expression for the volume in terms of c and the volume of P(x).
  • Changing the given volume does not make the problem more difficult.

Next week 2021 AB 3/ BC 3.

I would be happy to hear your ideas for other ways to use this questions. Please use the reply box below to share your ideas.



Area and Volume Problems (Type 4)

AP Type Questions 4: Area and Volume

Given equations that define a region in the plane students are asked to find its area, the volume of the solid formed when the region is revolved around a line, and/or the region is used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every year since year one (1969) on the AB exam and almost every year on the BC exam. You can be pretty sure that if a free-response question on areas and volumes does not appear, the topic will be tested on the multiple-choice section.

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). Students should know how to store and recall these values to save time and avoid copy errors.
  • 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 or an edge of the region, by the disk/washer method. See “Subtract the Hole from the Whole”
  • The cylindrical shell method will never be necessary for a question on the AP exams, 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. For an interesting variation on this idea 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 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: A=\tfrac{1}{2}\int\limits_{{{\theta }_{1}}}^{{{\theta }_{2}}}{{{\left( r\left( \theta  \right) \right)}^{2}}}d\theta
  • For BC only – Find perimeter using arc length integral

If this question 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 a student gives 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 in this case the integrals will be easy or they will be set-up-but-do-not-integrate questions.)

Occasionally, other type questions have been included as a part of this question. See 2016 AB5/BC5 which included an average value question and a related rate question along with finding the volume.

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 911, 2013 and “Subtract the Hole from the Whole” of December 6, 2016.


The Area and Volume question covers topics from Unit 6 of the 2019 CED .


Free-response questions:

  • Variations: 2009 AB 4, Don’t overlook this one, especially part (b)
  • 2016 AB5/BC5,
  • 2017 AB 1 (using a table),
  • 2018 AB 5 – average rate of change, L’Hospital’s Rule
  • 2019 AB 5
  • Perimeter parametric curves 2011 BC 3 and 2014 BC 5
  • Area in polar form 2017 BC 5, 2018 BC 5, 20129 BC 2

Multiple-choice questions from non-secure exams:

  • 2008 AB 83 (Use absolute value),
  • 2012 AB 10, 92
  • 2012 BC 87, 92 (Polar area)

 

 

 

 

 

 

Revised March 12, 2012


 

2019 CED Unit 8: Applications of Integration

This unit seems to fit more logically after the opening unit on integration (Unit 6). The Course and Exam Description (CED) places Unit 7 Differential Equations before Unit 8 probably because the previous unit ended with techniques of antidifferentiation. My guess is that many teachers will teach Unit 8: Applications of Integration immediately after Unit 6 and before Unit 7: Differential Equations. The order is up to you. 

Unit 8 includes some standard problems solvable by integration (CED – 2019 p. 143 – 161). These topics account for about 10 – 15% of questions on the AB exam and 6 – 9% of the BC questions.

Topics 8.1 – 8.3 Average Value and Accumulation

Topic 8.1 Finding the Average Value of a Function on an Interval Be sure to distinguish between average value of a function on an interval, average rate of change on an interval and the mean value

Topic 8.2 Connecting Position, Velocity, and Acceleration of Functions using Integrals Distinguish between displacement (= integral of velocity) and total distance traveled (= integral of speed)

Topic 8. 3 Using Accumulation Functions and Definite Integrals in Applied Contexts The integral of a rate of change equals the net amount of change. A really big idea and one that is tested on all the exams. So, if you are asked for an amount, look around for a rate to integrate.

Topics 8.4 – 8.6 Area

Topic 8.4 Finding the Area Between Curves Expressed as Functions of x

Topic 8.5 Finding the Area Between Curves Expressed as Functions of y

Topic 8.6 Finding the Area Between Curves That Intersect at More Than Two Points Use two or more integrals or integrate the absolute value of the difference of the two functions. The latter is especially useful when do the computation of a graphing calculator.

Topics 8.7 – 8.12 Volume

Topic 8.7 Volumes with Cross Sections: Squares and Rectangles

Topic 8.8 Volumes with Cross Sections: Triangles and Semicircles

Topic 8.9 Volume with Disk Method: Revolving around the x– or y-Axis Volumes of revolution are volumes with circular cross sections, so this continues the previous two topics.

Topic 8.10 Volume with Disk Method: Revolving Around Other Axes

Topic 8.11 Volume with Washer Method: Revolving Around the x– or y-Axis See Subtract the Hole from the Whole for an easier way to remember how to do these problems.

Topic 8.12 Volume with Washer Method: Revolving Around Other Axes. See Subtract the Hole from the Whole for an easier way to remember how to do these problems.

Topic 8.13  Arc Length BC Only

Topic 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled  BC ONLY


Timing

The suggested time for Unit 8 is  19 – 20 classes for AB and 13 – 14 for BC of 40 – 50-minute class periods, this includes time for testing etc.


Previous posts on these topics for both AB and BC include:

Average Value and Accumulation

Average Value of a Function and Average Value of a Function

Half-full or Half-empty

Accumulation: Need an Amount?

AP Accumulation Questions

Good Question 7 – 2009 AB 3 Accumulation, explain the meaning of an integral in context, unit analysis

Good Question 8 – or Not Unit analysis

Graphing with Accumulation 1 Seeing increasing and decreasing through integration

Graphing with Accumulation 2 Seeing concavity through integration

Area

Area Between Curves

Under is a Long Way Down  Avoiding “negative area.”

Improper Integrals and Proper Areas  BC Topic

Math vs. the “Real World”  Improper integrals  BC Topic

Volume

Volumes of Solids with Regular Cross-sections

Volumes of Revolution

Why You Never Need Cylindrical Shells

Visualizing Solid Figures 1

Visualizing Solid Figures 2

Visualizing Solid Figures 3

Visualizing Solid Figures 4

Visualizing Solid Figures 5

Painting a Point

Subtract the Hole from the Whole and Does Simplifying Make Things Simpler?

Other Applications of Integrals

Density Functions have been tested in the past, but are not specifically listed on the CED then or now.

Who’d a Thunk It? Some integration problems suitable for graphing calculator solution


Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations  Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration   Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions 

2019 CED Unit 10 Infinite Sequences and Series


 

 

 

 

 

 

Type 4 Questions: Area and Volume Problems

Given equations that define a region in the plane students are asked to find its area, the volume of the solid formed when the region is revolved around a line, and/or the region is used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every year since year one (1969) on the AB exam and almost every year on the BC exam. You can be pretty sure that if a free-response question on areas and volumes does not appear, the topic will be tested on the multiple-choice section.

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). Students should know how to store and recall these values to save time and avoid copy errors.
  • 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 or an edge of the region, by the disk/washer method.
  • The cylindrical shell method will never be necessary for a question on the AP exams, 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. For an interesting variation on this idea 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 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: A=\tfrac{1}{2}\int\limits_{{{\theta }_{1}}}^{{{\theta }_{2}}}{{{\left( r\left( \theta  \right) \right)}^{2}}}d\theta
  • For BC only – Find perimeter using arc length integral

If this question 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 a student gives 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 in this case the integrals will be easy or they will be set-up-but-do-not-integrate questions.)

Occasionally, other type questions have been included as a part of this question. See 2016 AB5/BC5 which included an average value question and a related rate question along with finding the volume.

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 911, 2013 and “Subtract the Hole from the Whole” of December 6, 2016


Free-response questions:

  • 2014 AB 2, 2013 AB 5.
  • 2015 AB 2
  • Variations: 2009 AB 4,
  • 2016 AB5/BC5,
  • 2017 AB 1 (using a table),
  • Perimeter 2011 BC 3 and 2014 BC 5

Multiple-choice questions from non-secure exams:

  • 2008 AB 83 (Use absolute value),
  • 2012 AB 10, 92
  • 2012 BC 87, 92 (Polar area)

 

 

 

 

 

 


Revised to add perimeter question 3-16-18,

Revised March 12, 2021

Applications of Integration – Volume

One of the major applications of integration is to find the volumes of various solid figures.

Volume of Solids with Regular Cross-sections  This is where to start with volume problems. After all, solids of revolution are just a special case of solids with regular cross-sections.

Volumes of Revolution

Subtract the Hole from the Whole and Does Simplifying Make Things Simpler?

Visualizing Solid Figures

Why you Never Need Cylindrical Shells

Painting a Point


 

 

 

 

 

Revised and update October 22, 2018