This is a free math program from Microsoft that is actually very handy to have. Just go to www.microsoft.com and type in “Microsoft Math” in the search window and you’ll be shown where to go. Completely free from Microsoft, available for immediate download..
Even though I have the very pricey MathCad software program, I still use this simple software to solve basic integration problems when trying to find the area under a curve and I’ll give examples of doing that further down the page. First, though, take a look at Figure 1 to see how the program looks when you first open it.
Figure 1: Microsoft Math Opening Screen
In Figure 2, I show an example of doing a linear regression in Excel, fitting a line to five different oxygen uptake rate (OUR) test results, conducted over a time period of about 250 minutes. The first OUR test was run four minutes after being brought to the lab and the last sample was tested after 241 minutes. The “sample” is a mixed liqour suspended solids (MLSS) sample. This same sample was aerated for the entire test period and was used for each of the five OUR tests graphed in Figure 2. Note that the “strength of the relationship”, the R2 value, a measure of how well the “oxygen demand” on the y-axis is predicted by the “time under aeration” on the x-axis, is 82.6% which is very good. But we can probably improve the R2 by using a different line fit to the data and we’ll do this in just a moment.
Figure 2: Simple Linear Regression in Excel
If we calculate the area under the curve, which is the straight, dashed line in the figure, we would be calculating the total oxygen consumption from time = 4 minutes to time = 241 minutes. One way to calculate the area, if you don’t know calculus, would be to use simple geometry to calculate the area in the rectangle plus the area in the triangle, as depicted in Figure 3. But this is cumbersome, to say the least. So let’s use Micrsoft Math to do the integration we need. First we’ll run another regression using a polynomial to obtain a better fit to the data. Then we’ll integrate the regression equation in Figure 3 as well as the new equation shown in Figure 4.
Figure 3: Using Geometry to Estimate the Area Under the Curve
In Figure 4 we’ve fit a new regression line using a 2nd order polynomial. The fit is much better as evidenced by the R2 value which is now 91.3%. In Figure 5 you can see the integration results for the two equations as calculated using Microsoft Math.
Figure 4: New Regression Line with Better Fit
We’ve done two regressions, a linear and a polynomial, and we’ve seen that the polynomial provides a better fit to the data. We’ve also used each regression to calculate the area under the curve to estimate the total oxygen consumption over a period of (241 - 4) = 237 minutes, the period of time our MLSS sample was being aerated while we ran the periodic OUR tests.
In Figure 5 we’ve used Microsoft Math to integrate our two regression equations. The integration of the linear regression shows that 2,913 mg/L O2 were consumed by the bacteria during the aeration period. The integration of the polynomial regression shows that 2,775 mg/L O2 were consumed by the bacteria during the aeration period. Our confidence in the accuracy of the linear regression was 83% and our confidence in the polynomial regression is 91%. From this laboratory testing we could extrapolate oxygen consumption in a full-scale biological reactor and we would have a high degree of confidence in our estimate.
Figure 5: Integration Results Using Microsoft Math
There’s one more nice feature about Micrsoft Math I want to show. In Figure 4 we used MS Math to do two integrations. But all we see are the equations and the results. We don’t know the individual steps taken by MS Math in doing the integrations. So, if we want to learn how integration is done, we’re left having learned nothing at this point. And that’s the final feature I want you to see, in Figure 6. You just click the “Solution steps” button in MS Math and the details of how the integration is done are shown.
Figure 6: Step-by-Step Integration Using Microsoft Math