Posts Tagged ‘Trigonometric Substitution’

Integration: When “u-Substitution” Fails

Posted: March 10, 2016 in Calculus: An Introduction, Integral Calculus, Trigonometric Substitution
Tags: , ,
0

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.

Arc Length

Posted: February 21, 2016 in Arc Length, Calculus: An Introduction, Differential Equations, Integral Calculus, Trigonometric Substitution
Tags: , ,
0

Surface areas of curved 3-dimensional solids tend to be much more difficult for students to conceptualize than those whose sides do not stray from a “level” plane. These will eventually be addressed but we will first discover how to calculate lengths of curves.

The circle will once again be called upon to initiate this exploration; the image below illustrates, in part, the method of exhaustion that Archimedes utilized to arrive at his estimate for π.

Archimedes Circumference2

Click on the link here to interact with what Archimedes revealed.

 

……..and now this. Was Archimedes wrong???

IMG_5215

Source: math.stackexchange.com

If the fellow above had joined pairs of points at each successive corner with a line segment (hypotenuse) and based his calculation for circumference on the sum of those, he would have found that Archimedes was correct all along.

 

Arc Length Formula

As  Δs approaches zero in this exploration, its length becomes a more accurate estimate for the arc length near the “point” of tangency (there are always two points in very close proximity). The end result through the limiting process shown directly below is the formula for arc length.

ArcLength Circle

Its always beneficial for students to work through several examples to cement their understanding of new concepts and related procedures; my preference is to provide examples that are already familiar to them. Calculating arc length (in this case) can then serve as a verification and acceptance of the new concept is achieved with confidence. The offering directly below and the link that follows connects the arc length formula to the Pythagorean theorem.

Arc Length

The formula for arc length is based on the Pythagorean theorem; it is therefore not surprising that they produce the same lengths on linear functions.

 

The real power of the formula for arc length lies in its applications to curves. Since students have known the circle’s circumference for several years, it is appropriate to now derive 2πr using our new tool. This is shown below and once again brings trigonometric substitution into play.

Circumference of the Circle

Arc Length (Circumference of Circle)

Once the circle’s circumference has been established using the arc length formula, the integration process can be further solidified by using arc length to once again calculate the circle’s area.

 

The formula for arc length will once again be employed in deriving the formulas for surface area of the sphere and cone.

 

Thanks for reading.

 

Trigonometric- and u-substitution

Posted: February 17, 2016 in Area, Calculus: An Introduction, Integral Calculus, Trigonometric Substitution
Tags: , , ,
0

When working with composite functions, u-substitution often simplifies the integration process for us. What is sometimes overlooked, however, is the significant adjustment that occurs to the area beneath the curve.  The image and link below allow students to explore the significance of this procedure.

ChainRule Area Cos

Interactively explore u-substitution.

 

As seen below, u-substitution is not a viable approach to integration as its leads to division by zero. In the case below, and others like it, trigonometric substitution provides a means for us to move forward and solve many problems. The process below also brings trigonometric identities into play, thereby  exposing students to their significance.

Circle Area Trig Substitution Formula1

Circle Area Trig Substitution Formula2

Thanks for reading.