1. C2 Chapter 11 Integration Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 17 th October 2013

2. Recap ? ? ? ? ?

3. Definite Integration We could add together the area of individual strips, which we want to make as thin as possible…

4. Definite Integration

5.  Reflecting on above, do you think the following definite integrals would be positive or negative or 0?  + 0   + 0   +0

6. Evaluating Definite Integrals We use square brackets to say that we’ve integrated the function, but we’re yet to involve the limits 1 and 2. Then we find the difference when we sub in our limits. ? ? ?

7. Evaluating Definite Integrals ? ? Bro Tip: Be careful with your negatives, and use bracketing to avoid errors.

8. Exercise 11B 1 2 4 6 a c e ? ? ? ? ? ?

9. Harder Examples Sketch: (Hint: factorise!) ? ? ?

10. Harder Examples -3 1 The SketchThe number crunching ??

11. Exercise 11C 1 2 3 4 5 ? ? ? ? ?

12. Curves bound between two lines

13. How could we use a similar principle if we were looking for the area bound between two lines? ? therefore area… ?

14. Curves bound between two lines Bro Tip: We’ll need to find the points at which they intersect. ?

15. Curves bound between two lines Edexcel C2 May 2013 (Retracted) ? ?

16. y = x(x-3) y = 2x AB C More complex areas Bro Tip: Sometimes we can subtract areas from others. e.g. Here we could start with the area of the triangle OBC. ?

17. Exercise 11D 1 3 4 9 ? ? ? ?

18. (Probably more difficult than you’d see in an exam paper, but you never know…) Q6 7 7 ?

19. y1y1 y2y2 y3y3 y4y4 hhh Trapezium Rule Instead of infinitely thin rectangular strips, we might use trapeziums to approximate the area under the curve. What is the area here? ?

20. Trapezium Rule In general: width of each trapezium Area under curve is approximately x11.522.53 y12.2546.259 ? Example ?

21. 0.8571 ? ? Trapezium Rule May 2013 (Retracted)

22. To add: When do we underestimate and overestimate?