# SL Integration + Trapezium rule

Welcome to integration! A lot of fun is in store! It helps to remember that this topic, along with differentiation, is the best description humans have yet invented for measuring, describing and calculating the movement of . . **anything **(challenge: name something that doesn't move - given a long enough time period . . )!

The teaching slides, student worksheet and online quizzes on this page cover the syllabus items: SL5.5 and SL5.8, related to integration. However, some of the slides and worksheets refer to activities listed in the "Activities" section below which the class will need to access for the applets and activities contained on those pages.

The relevant teaching slides and student worksheets from below are reproduced on each activity page so that the activity pages can "stand alone" if required. This page effectively provides an overview of all sub-topics, and activities, required for 'integration'.

## Teaching slides

Please find below some slides to help present and teach these ideas to students. You can click on the fullscreen "Presentation mode" from the tools on the right-hand side of this page: to make the slide fill the screen. Keep clicking on the ARROW on the RIGHT until it is greyed out/it is no longer possible: to see all the detail on each slide.

## Practice questions

Displacement, Velocity and Time quick practice to test students understanding for calculating displacement(distance), velocity(speed) or time, given data on the other two.

Here are some Trapezium Rule - Exam Style Questions - 100 marks worth to put yourself to the test!

*teaching activities*listed below:

## Teaching Activities

The following is a list of teaching ideas that relate to the teaching of this topic. Some will be links to activity pages and some sill just be ideas. Teachers will be the first visitors to these pages, but it is anticipated that these will be given student access and can be set as assignments for students.

The applets/matching activities on this page relate every student's experience of movement to the shape of a velocity/speed-time graph. Students may have have already investigated distance-time graphs using activities such as: Distance Time Graphs and Travel Graphs.

This activity offers video footage, and the corresponding real-world data set, 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.

Students will investigate the areas under straight lines and curves using the applets available on this page.

This helps develop the key skill of investigation, the *scientific method*, **Experiment, record results, organise, look for pattern, hypothesise(conjecture), test the conjecture **. . . and for mathematicians, **prove ! **

Students learn the notation (symbosl/language) used for defining, and calculating, areas under curves (definite integrals) and anti-derivatives (indefinite integrals), then put their understanding to the test through games and questions.

## Unit Planning

**ToK****Toricelli's Trumpet Paradox (aka Gabriel's Horn)**: a shape with **infinite **surface area but **finite **volume. Overview. The mathematics.