Archive for the ‘Calculus: An Introduction’ Category

The power of integral calculus is once again exploited in this entry, this time to determine the centers of mass of one- and two-dimensional objects. Before getting to that, however, some preliminary “discourse” will be engaged in to set the stage.

Everyone (of my age at least) can relate to the scenario involving two children playing on a see-saw. If the children have equal mass and are sitting an equal distance from the fulcrum, they can achieve perfect balance; the fulcrum in this scene is located at the center of mass. If, however, two differing masses are involved (all other things remaining equal), the side containing the greater mass will rotate downward on the fulcrum. This brings us to a very important term for this and other concepts involving rotational motion. This term is called the MOMENT of a force and is defined below:

The moment of a force is a measure of its tendency to rotate about an axis or a point. The moment can be influenced by two quantities: the object’s mass and its distance from the axis or point of rotation; this distance can be referred to as the moment arm.

Moment=Force x Distance , with units measured in ft-lb, kgf-m, etc.

In our example, when two children of the same mass are positioned an equal distance on opposing sides of the fulcrum, a state of equilibrium is achieved. If a bird suddenly joins in the fun and lands on the head of one of the participants, the mass on that end is increased and begins to rotate downward; this increase in mass has created a moment. A moment can also be created if one participant increases his/her position relative to the fulcrum. The moment arm on that side has now been lengthened, thereby creating a moment.

 

When finding the center of mass in one dimension, the same principles apply; this is a very straight forward procedure if the object has uniform mass density over its entire length. Complications arise, however, if the object’s mass density is not uniform throughout. To address this issue, the object is analyzed as a collection of very small points, each having its own mass and unique position (moment arm) within the main object; each of these will be referred to as “point-mass”.  As in the playground scenario described earlier, the further each point-mass is situated from the axes (or point) of rotation, the greater its contribution will be to the moment and, thereby, its influence on the location of the center of mass. The calculation for center of mass is built upon the concept of weighted averages; while the most frequently occurring outcomes have a significant say with respect to the overall average, the extreme outliers can also have a measurable impact.

 

Before weighted averages can be referenced in this context, the notation and the underlying concept that will be utilized throughout must be introduced. This is initiated below in the Mean Value Theorem of integral calculus.

Mean Value Theorem of Integral Calculus

CenterofMassMean Value Theorem

NOTE: In the calculations that follow, x- and y-components of the moment appear. Since centers of mass occur at a point of equilibrium, force due to gravity is ignored and omitted from the units chosen to represent those quantities. I wanted clarification on this item here since moments are once again called upon when dealing with torque. In that application, force is included in the units of measurement when describing moments of inertia.

 

A reference was made earlier to “point-masses” and their relative position within the object containing them. A direct connection between this and weighted averages exists and is presented below.

Weighted Average

CenterofMassWeighted Average

 

The end result in the first two examples in the following image are common sense and serve as a “trial run” on the theory developed above; all three can be related directly to our playground scenario described earlier.

 

One Dimensional Center of Mass

Centers of Mass (examples)

 

While limiting ourselves here to one-dimension would be silly, attempting to illustrate centers of mass in three dimensions on a 2-D surface could be considered reckless. For this reason, two-dimensions will be the extent to which this topic is pursued here.

Two Dimensions: x-component

Centers of Mass (2D) x-component

 

Two Dimensions: y-component

Centers of Mass (2D) y component

 

In the examples that follow, centers of mass are determined using the theory developed above. Interactive links also provide the opportunity to change one or more parameters in these examples to observe variations in the various integrals involved.

Constant Function

CenterofMassConstant

Click on the link provided here to interact with centers of mass on one dimension.

 

Linear Function

Center of Mass example (2D)

 

One Image from the exploration that follows……

CenterofMassLinear

Click on the link provided here to explore centers of mass defined by linear functions.

 

Quadratic Function

CenterofMassQuadratic

Click on the link provided here to explore centers of mass defined by quadratic functions.

 

 

The examples that follow have mass density increasing exponentially along the x-axis. With exponentials in the mix, the need for a new method of integration (by parts) emerges; the power of WolframAlpha is also introduced to do the “heavy lifting”.

 

Constant Function: Exponential Increase in Mass Density

CenterofMassConstant2a

 

The x-component of the center of mass in the example above can be calculated manually using integration by parts; this procedure is included here.

Integration by Parts

IntegrationbyPartsCoM

 

Click on the link provided here to explore centers of mass resulting from an exponential increase in mass density.

 

 

While integration by parts can be exploited to evaluate all integrals of this form, the process can become a time-consuming one. The following example is one result obtained from the exploration directly above; it is included here with the intent of introducing students to the power of WolframAlpha. The x-component of the center of mass is shown in a screenshot of the WolframAlpha app available on any device. Interested students have the option of verifying this and other results manually using integration by parts.

 

Quadratic Function: Exponential Increase in Mass Density 

CenterofMassConstant2

 

WolframAlpha App

WolframAlpha

I’ve included a link here to the web-based version of the app shown above. To verify the y-component of the center of mass in the final example shown above, click on WolframAlpha.

 

Thanks for reading.

 

References

Courant, Richard., John, Fritz (1999).  Introduction to Calculus and Analytics: Classics of Mathematics. New York, NY:  Springer-Verlag Berlin Heidelberg.

Larson, R., Hostetler, R. P., & Edwards, B. H. (1995). Calculus of a Single Variable: Early Transcendental Functions. Lexington, MA: D.C. Heath.

Pressure is a force per unit of area exerted over the surface of an object (as in 35 psi in the tires on your car). When an object is immersed in water, or some other liquid medium, the fluid pressure exerted on that object varies with the depth at which it is submerged. For example, the volume of water pressing down on an object submerged 10 feet is twice that of an object submerged half that depth.

Fluid Pressure (force per unit area) can therefore be defined as follows:

p=wh ,where w = weight density of the fluid h = depth at which the object is submerged

Fluid Force (total force exerted on object) is therefore given by

F=pA ,where A=total area of surface object in question F=whA

According to Pascal’s Law (principle), an object submerged in a fluid is subjected to equal pressure in all directions (at any given depth). For a sheet of metal submerged in water and resting horizontally at a given depth, the fluid force is constant over its entire surface. If, however, the submerged sheet is resting vertically, the entire force exerted over this object by the water varies with depth; the bottom of the sheet will experience more fluid force than its top. In order to determine the total force acting on this vertically oriented sheet, the force exerted on each rectangular cross-section of infinitesimal width (Δy) will be calculated and summed over the object’s entire vertical span; enter integration. With a well-chosen location for the y-axis in our model, the length of each rectangular cross-section can be easily expressed as some variation of f(y).

 

Rectangular Plate

Fluid ForceRectangularPlate

 

The example above was relatively simple since f(y) was a constant throughout its vertical span. With shapes whose widths are not constant, the mathematical model can once again vary depending on the perspective chosen. In the examples below, circular plates have been introduced since they provide opportunities for multiple forms of substitution in the integration process, thereby maximizing learning opportunities for students.

 

The fluid force acting on the ends of a cylindrical water tank is the subject below. I felt that this would be more interesting than imagining the force exerted on a circular plate submerged in a body of water. To simplify matters here, atmospheric pressure and other factors such as sliding forces have been ignored.

Cylindrical Tank: Half-full (Perspective 1)

tank2

 

Cylindrical Tank: Half-full (Perspective 2)

tank

 

Cylindrical Tank: Filled to Capacity

tank3

 

It is worthwhile drawing attention to the forces acting on the ends of top half of the tank filled to capacity (451.34 lb) and the bottom half filled to half capacity (332.8 lb). This type of thoughtful comparison can add to the students’ understanding of this topic.

 

The two images directly below once again show the two perspectives of fluid forces acting on the ends of our water tank. They are included here to illustrate the contents of the interactive link that follows.

Tank Centered at (0,0)

CircularPlateSubmerged2

 

Tank Centered at (0,-2)

CircularPlateSubmerged2b

Click on the link provided here to interact with fluid force on the ends of a cylindrical tank.

 

In the image and link below, the circular end has been removed from the tank and submerged on its own. Once again, various mathematical models could be used to describe this scenario; the one chosen here has placed the origin at the circle’s center.

CircularPlateSubmerged3

Click on the link provided here to interact with fluid force on a submerged vertical plate with center at origin.

 

The following links will be of interest to some:

Pressure and Buoyancy and Hydrodynamics

 

Thanks for reading.

 

Reference
Larson, R., Hostetler, R. P., & Edwards, B. H. (1995). Calculus of a Single Variable: Early Transcendental Functions. Lexington, MA: D.C. Heath.

Here’s problem involving a falling object and the speed at which its shadow travels along the ground. As usual, in related rates, once a relationship between the variables involved has been established, the calculus required to reach its conclusion is very straight forward.

In order to make efficient use of time, these problems provide students the opportunity to practice simple differentiation procedures. In addition, the graphs provided below open the door to a discussion on the Mean Value Theorem of differential calculus, serving to either introduce or reinforce that concept.

 

Falling Ball

RelatedRatesBallDrop

Click on the link provided here to interact with the falling ball and its shadow.

 

 

The ball’s displacement from its release point was provided in the image above. As a review (since integral calculus has already been introduced), that displacement formula is once again derived through basic differential equations; this is shown directly below.

Acceleration, Velocity and Displacement

BallDropDisplacementFormula

 

I’ve included solutions for  t=1  and  t=2  below. In keeping with my belief that students can learn effectively through comparison and contrast, three varied methods are shown.

Solutions

RelatedRatesBallDropSolutions

Thanks for reading.

The main subject of this entry was originally planned as an optimization problem involving differential calculus only; its been slightly modified. This more interesting approach provides the derivative up front, presenting students with three separate tasks to pursue from that point. As a consequence, students are reintroduced to differential equations and curve sketching.

A talking point emerges as well: Is there a difference between derivatives and differential equations?

 

Inscribed Triangle of Maximum Area

Inscribed Triangle animation

 

Click on the link provided here to explore area of the inscribed triangle.

 

Thanks for looking.

As mentioned earlier, my students’ first exposure to calculus is from the “Leibniz” perspective. Introducing integration in the first two weeks opens the door to the exploration of many interesting scenarios, including separable differential equations. These lead very nicely to discussions on rates of change,  providing a seamless segue into the land of differential calculus and derivatives.

After students have a firm hold on the concept of differential calculus, I like to shake things up once in a while by throwing in some integration problems. The subject of this entry, shown below, is one such problem. Students are asked to determine the area bounded by the function  f(x)=4/(1+x^2) and the x-axis from  x=0  to  x=1. When the result emerges, students are highly motivated to understand what’s going on. A great discussion ensues and a plan of attack is formulated; this first attempt is a good one but it eventually ends in a stalemate as illustrated below.

 

Why u-substitution fails

Pi Estimate Natural Log Attempt

 

Allowing students to travel down the wrong path often leads to greater learning than might otherwise occur; the example above is no exception. Learning various methods of integration is one thing, but knowing when to employ one approach over another is very empowering as well.

The impasse that was reached above provides motivation to seek out another approach; enter trigonometric substitution. This is a very powerful tool in the arsenal once students learn to recognize the circumstances in which it can be used. It is demonstrated below and leads to the conclusion of this problem.

 

Trigonometric Substitution

Pi Estimate Trig Sub

Other examples of trigonometric substitution can be seen in calculating  circumference  and area of the circle.

Later on this semester, students will be challenged to determined the arc length on the function  f(x)=x^2  over a given interval; trigonometric substitution will once again be deployed. This time, however, the solution will not be determined so easily. The roadblock in this scenario will lead to even greater learning, hyperbolic trig-substitution included.

 

Thanks for reading.

As the title indicates, telescoping sums and mathematical induction are the focus of this entry. Having said that, since the formula for sum of triangular numbers is required in the telescoping sum chosen, I’ve included one version of its derivation here as well.

The method shown below for the sum of triangular numbers involves duplicating the original “stack of blocks”; this duplicate stack is then rotated and affixed to the original stack to form a rectangle. Since two identical stacks have been combined to form this rectangle, its area must be halved to arrive at the desired result.

 

Sum of Triangular Numbers

Sum of Triangular Numbers2

Click on the link provided here to complete the rectangle shown above.

 

Mathematical Induction

When using induction to prove a mathematical statement to be true, the statement itself must first be given. A “base case” is shown initially to prove that the statement holds true for our starting point; this is usually either “0” or “1”.

The inductive proof for the sum formula of triangular numbers follows here. Once the statement is proven true for the basis, it is assumed that a value “k” will also hold true. If it can be shown that the statement also holds for a value “k+1”, then it can be assumed that it will hold true for all values of  k≥1.

 

Induction Proof (Sum of Triangular Numbers)

Induction (Sum of Triangular Numbers)

 

 

The formula for sum of squares is derived directly below using telescoping sums. Mathematical induction follows to prove that this formula holds true for all values of the variable.

 

Telescoping Sum (Sum of Squares Formula)

Telescoping Sum

 

Proof by Induction (Sum of Squares)

Induction

Induction2

Other methods are available for deriving each of the sum formulas shown here; another such entry (using linear combinations) will eventually be linked here. Regardless of how each formula is derived, mathematical induction is a very useful tool to test its reliability.

 

Thanks for reading.

 

With “Pi day” fast approaching, the timing of this is appropriate. One activity on day one with my introductory calculus students has them throwing toothpicks onto a large sheet of paper containing several strategically spaced parallel lines. After an adequate number of trials, students double that quantity, then divide that result by the number of toothpicks that touched or crossed any line. Once it is determined that the results are always relatively close to π, they want to know why.

This, along with some other discussions and demonstrations, set the stage very nicely for our little journey through introductory calculus. The entry here shows how part of the curiosity initiated on day one will be satisfied.

 

Buffons’ Needle (setup)

Buffon's Needle

 

Function Reflects the Moving Needle

Buffon's Needle2

 

Conclusion

Buffon's Needle4

 

I’ve included additional notes below to set up the interactive link that follows.

No Hit (wrong combination)

Buffon's Needle3Optional1

 

Point of Rotation on Line (always intersecting)

Buffon's Needle3Optional

Click on the link provided here to interact with Buffon’s Needle.

 

Thanks for reading.

 

Reference: http://mste.illinois.edu/activity/buffon/