# Numerical (GDC) integration 'Often there is NO analytical solution. Numerical methods are needed''

In science, and particularly in engineering, where mathematics has to be applied in a complex world, there is often no exact mathematical answer/solution. In such cases, repeated estimations and testing are used until a solution is found that is "good/safe enough".

Below are some real-world data sets that an engineer may well have to collect, but that don't seem to fit the range of mathematical functions learned in high school. This is where the area approximation formulae, using our calculators, can help find a "good/safe enough" solution.

### Resources

#### Practice Questions Numerical Integration Quiz questions using the Airbus plane data and the Downhill skiiing data below.

#### Airbus aircraft speed: take off to 10 000 feet student sheet/notes paper version of the below activities and questions + teacher presentation slides . . are available from the "Activities" section of the SL integration + trapezium page.The data below shows the velocity at take-off of an Airbus aircraft.

Task1: When no exact, analytical solution is possible, numerical methods can be used

We want to work out the total distance travelled by the aircraft = area under this velocity-time graph.

• Copy and paste this data into some graphing software (such as Desmos, Geogebra etc.).
• Use the software regression functions to try and find as accurate a model as you can. Take screenshots/save the files for the three best function models you have found.

It is hard to find a function that fits this data. This is common in engineering. Numerical methods/approximations, rather than exact mathematical analysis, are needed.

• Use your calculator and the formula given in the IBDP Mathematics Applications and interpretation formula booklet to work out the area under the curve = displacement, for the plane from 60 ≤ t < 80 seconds.
• Now use your calculator and Formula booklet trapezoidal rule to answer the questions shown below in the Practice Questions section (to view the solutions click on the 'eye' icon: )

#### Downhill skiing data

The data below shows the velocity of a downhill skier (you may be interested in researching the "fastest non-motorised humans on the planet"!).

We want to work out the total distance travelled by the skier = area under this velocity-time graph.

• Copy and paste this data into some graphing software (such as Desmos, Geogebra etc.).
• Use the software regression functions to try and find as accurate a model as possible.

As we found with the previous, Airbus airplane data, it is hard to find a function to fit the data. At HL, "piecewise" functions are introduced, which can help in such situations to develop an analytical model. In the absence of a suitable function model, we can also use the trapezoid rule numerical method (see your formula booklet).

If you need the extra practice, use the formula now to work out the total distance travelled . . . otherwise, your teacher may ask you to attempt, instead, the questions on the downhill skiing data found above.

SL5.5 syllabus number with topic summary

### Description

• These datasets have been collected in real-time from real experiments. A lot of data used in engineering does not perfectly fit a given function/analytical method. Numerical methods, such as the trapezoid rule used here, allow, in such situations, for solutions that are "good/safe enough". To bring this to life, students first have to experience for themselves, by copying and pasting the data into a graphing software package, the difficulty in finding an accurate analytical model = Task 1 and Task 2 above.

• Once they have finished trying to model the two data sets (and come up, hopefully, with some options that could work ok) they should attempt the practice questions above.

• The question on the downhill skiing data: "Between which two, consecutive, 5 second intervals is the change in the distance covered the greatest ?" is meant to hint at the concept of "acceleration" = gradient of the velocity-time graph. Acceleration and kinematics are the HL syllabus only, but could form part of a good SL internal assessment. It can also be used as further practice for using the trapezoid rule for calculating area. To avoid repetitive calculations (unless the repeated practice would help the class get comfortable with substituting the appropriate numbers into the formula on their GDCs) students could, in groups, each take charge of one five second period, and then share the results with the rest of the class (checking each other within their group to ensure no errors).

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