Posts Tagged ‘Integration’

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.

 

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As in the case of the cone, volume of the pyramid can be determined in various ways. The version here once again dabbles with a multi-variable scenario to give students some variation on a familiar theme.

Pyramid Volume2

Click on the link to interact with the notion of infinitesimals as applied to pyramid volume.

Refer to volume of cone for helpful comparison.

 

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Now that the formula for arc length has been determined, we can pursue surface areas of curved solids.

Some students have great difficulty conceptualizing areas on curved surfaces. The problem lies in the fact that they want to use Δx or Δy in the setup, just as they did when calculating volumes of solids and areas over flat surfaces. Since on a curved surface, both x and y are changing in unison; as one changes, so does the other. For this reason, we need to use a variable linking those two variables together. This brings into play the Pythagorean Theorem and arc length. Arc length is a one-dimensional measure, its formula the result of capturing the interplay between Δx and Δy and expressing that as Δs. Integrating the one-dimensional Δs over a given interval will therefore produce the desired outcome (even though the path might not be linear, its distance is nevertheless one-dimensional).

A similar argument can be made for calculating areas over curved surfaces. Since area is two-dimensional, the integral we set up to calculate area must stick to that. I visualize a sphere wrapped with very narrow “bands”. If one of these bands was removed and cut, it could be stretched out and laid flat; its length on one edge would exceed that of the opposing edge due to the fact that it was wrapped around a curved surface. Ultimately this is not a problem, however, since these two opposing edges approach the same length as the distance between them narrows. This is entirely similar to each annulus in a previous post where circle area was derived using the “onion proof“.

This takes care of one dimension required for surface area. The second dimension is arc length mentioned in paragraph two above. As  Δs approaches zero, the line segment joining the two infinitely close points that determine the “point” of tangency becomes ever more perpendicular to the length of each band mentioned earlier. Each band can be treated as a rectangle; area is determined as the product of width (arc length) and length (circumference of 3-D solid) at each x_i over the given interval.

 

The following notes reveal surface area of a sphere using the reasoning described above.

Sphere

Surface Area (sphere)Conventional

Click on the link to view the changing width of each “band” around the sphere.

 

As before, students can once again benefit having a second example from which to draw comparisons to the first; the cone serves this purpose very well.

Lateral Surface Area of Cone

Cone Lateral Area

 

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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.

 

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This entry will no doubt get further under the skin of those who contend that I’m pushing things too far (and that is exactly why I’m publishing it).  This is a natural extension of the 2-dimensional analog in which circle area was derived. Cylindrical coordinates are identical to polar coordinates with a 3rd dimension thrown in. Directly below, the vertical dimension is governed by a linear function which results in a cone.

In the spirit of consistency, I’ve included a second example with which to draw comparisons to the first. Everything is identical between the two cases with one exception; the function which governs the vertical dimension in example two is that of a semi-circle.

 

Volume of a Cone

VolumeCone Cylindrical

Interact with cylindrical coordinates to see how the 3-dimensional sector changes.

 

Volume of a Sphere

VolumeSphere Cylindrical

 

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Polar coordinates are not formally introduced in high school, yet sector area is required to prove fundamental trigonometric limits. Having already seen that proof, it wouldn’t be all that painful for students to see circle area derived through the limiting process as applied to angles.

Variable Angle Fixed Radius

Circle AreaPolar

 

I’ve included another scenario below, this time with variable radius as well.

Variable Angle, Variable Radius

 

Circle AreaPolar(delta r)

Explore changing polar coordinates here.

Connect your understanding of polar coordinates to that of cylindrical coordinates.

 

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Displayed below are two familiar methods of determining volumes of solids, this time as applied to the sphere. It is beneficial to compare two (or more) methods of determining volumes of a given solid; this allows students to develop a deeper understanding of each process that might not otherwise occur if kept in isolation from one another. It is also beneficial to compare directly each method as applied to different shapes. I’m including a link to volume of cones here for that purpose.

Volume SphereShells

Volume of Sphere (slabs)

See also volume of cones.

Thanks for reading.