Two ways of doing an integral (Part 1)

A colleague placed the following problem on an exam:

$\displaystyle \int \frac{dx}{\sqrt{4x-x^2}}$

He expected students to solve this problem by the standard technique, completing the square:

$\displaystyle \int \frac{dx}{\sqrt{4x-x^2}} = \displaystyle \int \frac{dx}{4-(x-2)^2} = \sin^{-1} \left( \displaystyle \frac{x-2}{2} \right) + C$

However, one student solved this problem by some clever algebra and the substitution $u = \sqrt{x}$, so that $x = u^2$ and $dx = 2u \, du = 2 \sqrt{x} \, du$:

$\displaystyle \int \frac{dx}{\sqrt{4x-x^2}} = \displaystyle \int \frac{dx}{\sqrt{x} \sqrt{4-x}}$

$= \displaystyle \int \frac{dx}{\sqrt{x} \sqrt{4 - (\sqrt{x})^2}}$

$= \displaystyle \int \frac{2 \, du}{\sqrt{4 - u^2}}$

$= 2 \sin^{-1} \left( \displaystyle \frac{u}{2} \right) + C$

$= 2 \sin^{-1} \left( \displaystyle \frac{\sqrt{x}}{2} \right) + C$

As he couldn’t find a mistake in the student’s work, he assumed that the two expressions were equivalent. Indeed, he differentiated the student’s work to make sure it was right. But he couldn’t immediately see, using elementary reasoning, why they were equivalent. So he walked across the hall to my office to ask me if I could help.

After a few minutes, I was able to show that the two expressions were equivalent.

I’ll leave this one as a cliff-hanger for now. In tomorrow’s post, I’ll show why they’re equivalent.

Engaging students: Completing the square

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Claire McMahon. Her topic, from Algebra: completing the square.

There were a lot of famous mathematicians that contributed to the notion of completing the square.  The first of the mathematicians was that of the Babylonians.  This culture started the notion of not only solving the quadratics but of arithmetic itself.  The Babylonians started with the equations and then proceeded to solve them algebraically.  Back then; they used pre-calculated tables to help them with solving for the roots.  They were basically solving by the quadratic equation at this point.  The man that came along has a very hard name to not only pronounce but to spell, and I will do my best.  I will refer to him as Muhammad from here on out but his full name, or one of the common names to which he is referred is Muhammad ibn Musa al-Khwarizmi.  He developed the term algorithm, which led to an algorithm for solving quadratic equations, namely completing the square.

The notion of completing the square has gone through a series of transformations throughout the history of mathematics.  As mentioned before the Babylonians started with the notion and increased the knowledge by developing the quadratic formula to find the roots of a given quadratic equation.  This spurred the thought that I can solve any equation and find its solution and roots by completing the square.  Muhammad brought this notion to us, of which was mentioned before.  More specifically the text that he developed was “The Compendious Book of Calculations by Completion and Balancing.”  This book of course has been translated several times over but the general idea is laid out in the title.  Modern mathematicians have developed a less compendious form that is now being taught in the math classes today.  They take on many different forms and can be taught with manipulates as well.

The fabulous part to the story is there are a lot of resources that help the kids of today to deal with this “trick” of the math trade.  There are numerous You Tube videos on the different methods of which show every step along the way with encouraging thoughts.  Another great online resource is any of the math websites.  I find it a little unfair that these resources were not readily available when I was struggling with such concepts.  One of my personal favorites is the PurpleMath.com website.  This website breaks everything down to basically that of a fourth grade level.  They have pictures and fun problems to work out on your own.  My favorite part is that you get your answers checked instantaneously to build the self-confidence and self-efficacy it takes to be a successful student.  These particular websites are great tools for teachers as well, as they have a lot of great examples that can be used in the classroom and different ways that a student might present and calculate a problem.