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C2 Chapter 11 Integration Dr J Frost Last modified: 17 th October 2013.

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Presentation on theme: "C2 Chapter 11 Integration Dr J Frost Last modified: 17 th October 2013."— Presentation transcript:

1 C2 Chapter 11 Integration Dr J Frost 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 a c e ? ? ? ? ? ?

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

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

11 Exercise 11C ? ? ? ? ?

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 ? ? ? ?

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 x y ? Example ?

21 ? ? Trapezium Rule May 2013 (Retracted)

22 To add: When do we underestimate and overestimate?


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