# Visualizing Unit 9

As you probably realize by now, I think graphs, drawing and other visuals are a great aid in teaching and learning mathematics. Desmos is a free graphing app that many teachers and students use to graph and make other illustrations. Demonstrations can be made in advance and shared with students and other teachers.

Recently, I was looking a some material from Unit 9 Parametric Equation, Polar Coordinates, and Vector-Valued Functions, BC topics, from the current AP Calculus CED. I ended up making three new Desmos illustrations for use in this unit. They will also be useful in a precalculus course introducing these topics. Hope you find them helpful.

Polar Graph Demo

You may replace the polar equation with any polar equation you are interested in. There are directions in the demo. Moving the “a” slider will show a ray rotating around the pole. The “a” value is the angle, $\displaystyle \theta$, in radians between the ray and the polar axis. On the ray is a segment with a point at its end. This segment’s length is $\displaystyle \left| {r\left( \theta \right)} \right|$. As you rotate the ray you can see the polar graph drawn. When $\displaystyle r(\theta )<0$ the segment extend in the opposite direction from the ray.

This demo may be used to introduce or review how polar equation work. An interesting extension is to enter something for the argument of the function that is not an integer muntiple of $\displaystyle \theta$ and extend the domain past $\displaystyle 2\pi$, for example $\displaystyle r=2+4\sin (1.2\theta )$

Basic Parametric and Vector Demo

A parametric equation and the vector equation of the same curve differ only in notation. So, this demo works for both. Following the directions in the demo, you may see the graph being drawn using the “a” slider. You may turn on (1) the position vector and its components, (2) add the velocity vector attached to the moving point and “pulling” it to its new position, and (3) the acceleration vector “pulling” the velocity vector.

You may enter any parametric/vector equation. When you do, you will also have to enter its first and second derivative. Follow the directions in the demonstration.

Cycloids and their vectors

This demo shows the path on a rolling wheel called a cycloid. The “a” slider moves the position of the point on the wheel. The point may be on the rim of the wheel ($\displaystyle a=r$, on the interior of the wheel ($\displaystyle a), or outside the wheel ($\displaystyle a>r$  – think the flange on a train wheel). Use the “u” slider to animate the drawing. The velocity and acceleration vectors are shown; they may be turned off. The velocity vector is tangent to the curve (not to the circle) and seems to “pull” the point along the curve. The acceleration vector “pulls” the velocity vector. The equation in this demo should not be changed.

The last two demonstrations give a good idea of how the velocity and acceleration vectors affect the movement of the point.

.

# Polar Equation Questions (Type 9)

### AP Questions Type 9: Polar Equations (BC Only)

Ideally, as with parametric and vector functions, polar curves should be introduced and covered thoroughly in a pre-calculus course. Questions on the BC exams have been concerned only with calculus ideas related to polar curves. Students have not been asked to know the names of the various curves (rose curves, limaçons, etc.). The graphs are usually given in the stem of the problem; students are expected to be able to determine which is which if more than one is given. Students should know how to graph polar curves on their calculator, and the simplest by hand. Intersection(s) of two graphs may be given or easy to find.

What students should know how to do:

• Calculate the coordinates of a point on the graph,
• Find the intersection of two graphs (e.g. to use as limits of integration).
• Find the area enclosed by a graph or graphs: $\displaystyle A=\frac{1}{2}\int_{{{{\theta }_{1}}}}^{{{{\theta }_{2}}}}{{{{{\left( {r\left( \theta \right)} \right)}}^{2}}d\theta }}$
• Use the formulas $\displaystyle x\left( \theta \right)=r\left( \theta \right)\cos \left( \theta \right)\text{ and }y\left( \theta \right)=r\left( \theta \right)\sin \left( \theta \right)$  to convert from polar to parametric form,
• Calculate $\displaystyle \frac{{dy}}{{d\theta }}$ and $\displaystyle \frac{{dx}}{{d\theta }}$ (Hint: use the product rule on the equations in the previous bullet).
• Discuss the motion of a particle moving on the graph by discussing the meaning of $\displaystyle \frac{{dr}}{{d\theta }}$ (motion towards or away from the pole), $\displaystyle \frac{{dr}}{{d\theta }}$ (motion in the vertical direction), and/or $\displaystyle \frac{{dx}}{{d\theta }}$ (motion in the horizontal direction).
• Find the slope at a point on the graph, $\displaystyle \frac{{dx}}{{dx}}=\frac{{dy/d\theta }}{{dx/d\theta }}$

When this topic appears on the free-response section of the exam there is no Parametric/vector motion question and vice versa. When not on the free-response section there are one or more multiple-choice questions on polar equations.

This question typically covers topics from Unit 9 of the CED.

Free-response questions:

• 2013 BC 2
• 2014 BC 2
• 2017 BC 2
• 2018 BC 5
• 2019 AB 2

Multiple-choice questions from non-secure exams:

• 2008 BC 26
• 2012 BC 26, 91

Other posts on Polar Equations

Polar Basics

Polar Equations for AP Calculus

Extreme Polar Conditions

Polar Equations (Review 2018)

Visualizing Unit 9 Desmos Demonstrations for Polar, Vector and Parametric Curves

Revised March 12, 2021, April 8, 2022

# 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 3/ BC 3
• 2022 AB2 – area, volume, inc/dec analysis, and related rate.
• 2023 BC 5 – Area, Improper integral, Integration by Parts (No area problem on 2023 AB exam)

Multiple-choice questions from non-secure exams:

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

Updated March 12, 2012, March 22, 2022, May 4, 2023, June 4, 2023

# Unit 9 – Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Unit 9 includes all the topics listed in the title. These are BC only topics (CED – 2019 p. 163 – 176). These topics account for about 11 – 12% of questions on the BC exam.

Comments on Prerequisites: In BC Calculus the work with parametric, vector, and polar equations is somewhat limited. I always hoped that students had studied these topics in detail in their precalculus classes and had more precalculus knowledge and experience with them than is required for the BC exam. This will help them in calculus, so see that they are included in your precalculus classes.

### Topics 9.1 – 9.3 Parametric Equations

Topic 9.1: Defining and Differentiation Parametric Equations. Finding dy/dx in terms of dy/dt and dx/dt

Topic 9.2: Second Derivatives of Parametric Equations. Finding the second derivative. See Implicit Differentiation of Parametric Equations this discusses the second derivative.

Topic 9.3: Finding Arc Lengths of Curves Given by Parametric Equations.

### Topics 9.4 – 9.6 Vector-Valued Functions and Motion in the plane

Topic 9.4 : Defining and Differentiating Vector-Valued Functions. Finding the second derivative. See this A Vector’s Derivatives which includes a note on second derivatives.

Topic 9.5: Integrating Vector-Valued Functions

Topic 9.6: Solving Motion Problems Using Parametric and Vector-Valued Functions. Position, Velocity, acceleration, speed, total distance traveled, and displacement extended to motion in the plane.

### Topics 9.7 – 9.9 Polar Equation and Area in Polar Form.

Topic 9.7: Defining Polar Coordinate and Differentiation in Polar Form. The derivatives and their meaning.

Topic 9.8: Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

Topic 9.9: Finding the Area of the Region Bounded by Two Polar Curves. Students should know how to find the intersections of polar curves to use for the limits of integration.

### Timing

The suggested time for Unit 9 is about 10 – 11 BC classes of 40 – 50-minutes, this includes time for testing etc.

### Previous posts on these topics:

Parametric and Vector Equations

Implicit Differentiation of Parametric Equations

A Vector’s Derivatives

Adapting 2012 BC 2 (A parametric equation question)

Polar Curves

Polar Equations for AP Calculus

Extreme Polar Conditions

Visualizing Unit 9 Desmos Demonstrations for Polar, Vector and Parametric Curves

# Extreme Polar Conditions

Looking at the graphs of polar curves can be quite fascinating. Doing “calculus” kinds of things is different, yet the same as in Cartesian coordinates. A discussion on the AP Calculus Community here got me thinking about extreme values of polar functions.

The terms maximum and minimum here refers to the value of r(θ) which may be positive (when on the ray making an angle of θ with the polar axis), zero (at the pole), or negative (on the ray opposite to the one making an angle of θ with the polar axis). The distance from the pole to the point is |r(θ)|.  As Mark Howell pointed out in the thread linked above, extreme values of r(θ) lie on a circle or circles centered at the pole with radius of |r(θ)|, and finding the slope of tangent lines at the extremes is relatively easy, requiring no calculus: the slope of the ray is y/x, so the slope of the tangent at the extreme is –x/y. As with Cartesian coordinates, at extreme values $\displaystyle dr/d\theta =0$ since r(θ) is change from increasing to decreasing at this point (or vice versa).

A quick look at the graph of simple polar functions seems to show obvious maximum values for r(θ), but a closer look reveals some complications.

The graph of $\displaystyle r\left( \theta \right)=\sin \left( {2\theta } \right)$, shown below, appears to show 4 maximums. However, if we trace the graph, we find that these points are (1, π/4), (–1, 3π/4), (1, 5π/4) and (–1, 7π/4). Two of the values are maximums where r(θ) = 1 and two are minimums where r(θ) = –1.

Thus, points may be both maximums and minimums.

Polar Curves can be really fun. While working on the idea above, I explored some other curves. Try some yourself using Desmos, GeoGabra, or another graphing app with sliders. Shown below are members of the family of polar curves $\displaystyle r\left( \theta \right)=A\cos \left( {B\theta } \right)+C\sin \left( {D\theta } \right).$ The domain is extended to $\displaystyle 0\le \theta \le 20\pi$. Notice:

• How very slight changes in the parameters give very different looking graphs
• Other values give far less “organized” curves
• In the third graph, the maximums and minimums on the irregular part of the curve closest to the pole

.

# Polar Equations for AP Calculus

A recent thread on the AP Calculus Community bulletin boards concerned polar equations. One teacher observed that her students do not have a very solid understanding of polar graphs when they get to calculus. I expect this is a common problem. While ideally the polar coordinate system should be a major topic in pre-calculus courses, this is sometimes not the case. Some classes may even omit the topic entirely. Getting accustomed to a new coordinate scheme and a different way of graphing is a challenge.  I remember not having that good an understanding myself when I entered college (where first-year calculus was a sophomore course). Seeing an animated version much later helped a lot.

This blog post will discuss the basics of polar equations and their graphs. It will not be as much as students should understand, but I hope the basics discussed here will be a help. There are also some suggestions for extending the study of polar function as the end.

Instead of using the Cartesian approach of giving every point in the plane a “name” by giving its distance from the y-axis and the x-axis as an ordered pair (x,y), polar coordinates name the point differently. Polar coordinates use the ordered pair (r, θ), where r, gives the distance of the point from the pole (the origin) as a function of θ, the angle that the ray from the pole (origin) to the point makes with the polar axis, (the positive half of the x-axis).

Start with this Desmos graph. It will help if you open it and follow along with the discussion below. The equation in the example is $\displaystyle r(\theta )=2+4\sin (\theta )$ You may change this to explore other graphs. (Because of the way Desmos graphs, you cannot have a slider for θ; the a-slider will move the line and the point on the graph. r(a) gives the value of r(θ).

• Notice that as the angle changes the point at varying distance from the pole traces a curve.
• Move the slider to π/6. Since sin(π/6) = 0.5, r(π/6) = 4. The red dot is at the point (4, π/6). Move the slider to other points to see how they work. For example, θ = π/2 gives the point (6,π/2).
• When the slider gets to θ = 7π/6, r = 0 and the point is at the pole. After this the values of r are negative, and the point is now on the ray opposite to the ray pointing into the third and fourth quadrants. The dashed line turns red to remind you of this.
• As we continue around, the point returns to the origin at θ = 11π/6, then values are again positive.
• The graph returns to its starting point when θ = 2π. Note (2,0) is the same point as (2, 2π).
• Even though this is the graph of a function, some points may be graphed more than once and the vertical line test does not apply.
• If we continued around, the graph will retrace the same path. This often happens when the polar function contains trig functions with integer multiples of θ.
• This does not usually happen if no trig functions are involved – try the spiral r = θ.
• If you enter non-integer multiples of θ and extend the domain to large values, vastly different graphs will appear, often making nice designs. Try $\displaystyle r\left( \theta \right)=2+4\sin \left( {1.3\theta } \right)$ for $\displaystyle 0\le \theta \le 20\pi$. This is an area for exploration (if you have time).

In pre-calculus courses several families of polar graphs are often studied and named. For example, there are cardioids, rose curves, spirals, limaçons, etc. The AP Exams do not refer to these names and students are not required to know the names. The exception is circles which have the following forms where R is the radius: θ=R, r = Rsin(θ) or r = Rsin(θ)

To change from polar to rectangular for use the equations $x=r\cos \left( \theta \right)$ and  $y=r\sin \left( \theta \right)$. This is simple right triangle trigonometry (draw a perpendicular from the point to the x-axis and from there to the pole).

To change from rectangular to polar form use  $r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}$ and  $\displaystyle \theta =\arctan \left( {\tfrac{y}{x}} \right)$

AP Calculus Applications

There are two applications that are listed on the AP Calculus Course and Exam Description: using and interpreting the derivative of polar curves (Unit 9.7) and finding the area enclosed by a polar curve(s) (Units 9.8 and 9.9).

Since calculus is concerned with motion, AP Students should be able to analyze polar curves for how things are changing:

• The rate of change of r away from or towards the pole is given by  $\displaystyle \frac{{dr}}{{d\theta }}$
• The rate of change of the point with respect to the x-direction is given by  $\displaystyle \frac{{dx}}{{d\theta }}$ where $\displaystyle x=r\cos \left( \theta \right)$ from above.
• The rate of change of the point with respect to the y-direction is given by  $\displaystyle \frac{{dy}}{{d\theta }}$ where $\displaystyle y=r\sin \left( \theta \right)$from above.
• The slope of the tangent line at a point on the curve is $\displaystyle \frac{{dy}}{{dx}}=\frac{{dy/d\theta }}{{dx/d\theta }}$. See 2018 BC5 (b)

Area

$\displaystyle \underset{{\Delta \theta \to 0}}{\mathop{{\lim }}}\,\sum\limits_{{i=1}}^{\infty }{{\tfrac{1}{2}}}{{r}_{i}}^{2}\Delta \theta =\tfrac{1}{2}\int_{{{{\theta }_{1}}}}^{{{{\theta }_{2}}}}{{{{r}^{2}}d\theta }}$

CAUTION: In using this formula, we need to be careful that the curve does not overlap itself. In the Desmos example, the smaller loop overlaps the larger loop; integrating from 0 to 2π counts the inner loop twice. Notice how this is handled by considering the limits of integration dividing the region into non-overlapping regions:

• The area of the outer loop is  $\displaystyle \tfrac{1}{2}\int_{{-\pi /6}}^{{7\pi /6}}{{{{{(2+4\sin (\theta ))}}^{2}}d\theta }}\approx 35.525$
• The area of the inner loop is  $\displaystyle \tfrac{1}{2}\int_{{7\pi /6}}^{{11\pi /6}}{{{{{(2+4\sin (\theta ))}}^{2}}d\theta }}\approx 2.174$
• Integrating over the entire domain gives the sum of these two:  $\displaystyle \tfrac{1}{2}\int_{0}^{{2\pi }}{{{{{(2+4\sin (\theta ))}}^{2}}d\theta }}=12\pi \approx 37.699$. This is not the correct area of either part.

This problem can be avoided by considering the geometry before setting up the integral: make sure the areas do not overlap. Restricting r to only non-negative values is often required by the fine print of the theorem in textbooks, but this restriction is not necessary when finding areas and makes it difficult to find, say, the area of the smaller inner loop of the example. Here is another example:

$\displaystyle r\left( \theta \right)=\cos \left( {3\theta } \right)$. Between 0 and $\displaystyle 2\pi$ this curve traces the same path twice.

# Parametric Equations, Polar Coordinates, and Vector-Valued Functions – Unit 9

Unit 9 includes all the topics listed in the title. These are BC only topics (CED – 2019 p. 163 – 176). These topics account for about 11 – 12% of questions on the BC exam.

Comments on Prerequisites: In BC Calculus the work with parametric, vector, and polar equations is somewhat limited. I always hoped that students had studied these topics in detail in their precalculus classes and had more precalculus knowledge and experience with them than is required for the BC exam. This will help them in calculus, so see that they are included in your precalculus classes.

### Topics 9.1 – 9.3 Parametric Equations

Topic 9.1: Defining and Differentiation Parametric Equations. Finding dy/dx in terms of dy/dt and dx/dt

Topic 9.2: Second Derivatives of Parametric Equations. Finding the second derivative. See Implicit Differentiation of Parametric Equations this discusses the second derivative.

Topic 9.3: Finding Arc Lengths of Curves Given by Parametric Equations.

### Topics 9.4 – 9.6 Vector-Valued Functions and Motion in the plane

Topic 9.4 : Defining and Differentiating Vector-Valued Functions. Finding the second derivative. See this A Vector’s Derivatives which includes a note on second derivatives.

Topic 9.5: Integrating Vector-Valued Functions

Topic 9.6: Solving Motion Problems Using Parametric and Vector-Valued Functions. Position, Velocity, acceleration, speed, total distance traveled, and displacement extended to motion in the plane.

### Topics 9.7 – 9.9 Polar Equation and Area in Polar Form.

Topic 9.7: Defining Polar Coordinate and Differentiation in Polar Form. The derivatives and their meaning.

Topic 9.8: Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

Topic 9.9: Finding the Area of the Region Bounded by Two Polar Curves. Students should know how to find the intersections of polar curves to use for the limits of integration.

### Timing

The suggested time for Unit 9 is about 10 – 11 BC classes of 40 – 50-minutes, this includes time for testing etc.

### Previous posts on these topics :

Parametric Equations

Vector Valued Functions

Polar Form