# What activities can enhance student comprehension of concepts involved in logarithmic and exponential integrals?

I will be teaching a calculus class, specifically, integration of common functions (e.g., polynomials, logarithms, exponentials and the like). It has been my experience that if an abstract concept can be at least related to a real life context, then it becomes easier for the students to comprehend and apply the particular concept.

What are effective real-life practical activities that will enhance student comprehension and ability to apply principles involved with logarithmic and exponential integrals?

• @Amaterasu: That seems like a very hasty reaction to me. We are only beginning to explore the scopes of this site and though many of us come from university teaching, this is likely to change. Mar 14 '14 at 9:25
• @Wrzlprmft quite a bit of meta discussion, this point I can understand and accept here
– user106
Mar 14 '14 at 9:38
• What makes those functions so special? Sure, they are "easy to handle" ones, but for the underlying concept of integral/area below the curve this is irrelevant. Mar 17 '14 at 17:09

In my answer to this question

https://math.stackexchange.com/questions/498339/demystify-integration-of-int-frac1x-dx/498790#498790

I show how the fact that the $$\ln(x) = \int_1^x \frac{1}{t}dt$$

can be understood as naturally in terms of trying to solve exponential equations.
You could convert this into a bonus assignment where they have to compute the solution to $3^x=2$ as a $16th$ century mathematician might: using only addition and multiplication. Let them make a spreadsheet or program, but require that they only use addition or multiplication in their programs. Then take them down the same path as in my story.

Although he teaches at high school level and it sounds like you're at university level, Shawn Cornally has great ideas for getting students to really experience the meaning of the calculus concepts they're working with. Here's one post on the natural logarithm: http://shawncornally.com/wordpress/?p=611

You can search his blog on 'exponential' to find lots more.

Practical examples may be found in various fields of science. Exponentially decreasing functions can be found in mechanics (a body falling in air, where the resistance is proportional to the velocity), electromagnetism (the charge on capacitor), chemical equilibrium (where the rates of reaction are proportional to the amount of reactants), biology (exponential growth and decay). More advanced logarithmic and exponential functions naturally appear when the expressions involved are not simple but depend on something else.

Since the integral of a postive function is the area under the function, a possible real-life application is the following situation:

Suppose you want to paint the wall of your room in a fancy manner. Namely, you want to draw a squiggly line (e.g the graph of $\frac{1}{x}$ or $\ln{x}$ in, say $[1,5]$) and paint the area below the line in red. How much red paint are you going to need for this?

• Can you rephrase your answer based on the edited question title?
– user89
Mar 14 '14 at 7:26
• @twirlobite: Can you clarify which kind of integrals you mean with logarithmic and exponential integrals? Mar 14 '14 at 7:29
• @Roland these web-formulas.com/Math_Formulas/…
– user106
Mar 14 '14 at 8:00
• @Amaterasu: I can include them in the answer, but I admit that this is a bit unsatisfactory. Mar 14 '14 at 8:07